?A rocket is fired straight up, burning fuel at the constant rate of \(b\) kilograms per

State the rule for integration by parts. In practice, how do you use it?

How do you evaluate \(\int \sin ^{m} x \cos ^{n} x \quad d x\) if \({m}\) is odd? What if \({n}\) is odd? What if \({m}\) and \({n}\) are both even? Equation Transcription: ? sinm x cosnx dx m n Text Transcription: integral sin^m x cos^n x dx m n

If the expression occurs in an integral, what substitution might you try? What if occurs? What if occurs?

What is the form of the partial fraction decomposition of a rational function if the degree of is less than the degree of and has only distinct linear factors? What if quadratic factor (not repeated)? What if the quadratic factor is repeated?

State the rules for approximating the definite integral with the Midpoint Rule, the Trapezoidal Rule, and Simpson's Rule. Which would you expect to give the best estimate? How do you approximate the error for each rule?

Define the following improper integrals. (a) (b) (c)

Define the improper integral for each of the following cases. (a) has an infinite discontinuity at . (b) has an infinite discontinuity at . (c) has an infinite discontinuity at , where .

State the Comparison Theorem for improper integrals.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\int \tan ^{-1} x \ d x\) can be evaluated using integration by parts

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. can be evaluated by applying integration by parts five times.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. To evaluate \(\int \frac{d x}{\sqrt{25+x^{2}}}\) an appropriate trigonometric substitution is \(x=5 \sin \theta\). Equation Transcription: ? x = 5 sin Text Transcription: integral dx/sqrt 25+x^2 x = 5 sin theta

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. To evaluate \(\int \frac{d x}{\sqrt{9+e^{2 x}}}\) we can use the formula in entry 25 of the Table of Integrals to obtain \(\ln \left(e^{x}+\sqrt{9+e^{2 x}}\right)+C\). Equation Transcription: ? ln() + C Text Transcription: integral dx/sqrt 9+e^2x ln(e^x+sqrt 9+e^2x)+C

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\frac{x\left(x^{2}+4\right)}{x^{2}-4}\) can be put in the form \(\frac{A}{x+2}+\frac{B}{x-2}\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. can be put in the form .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. can be put in the form

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. can be put in the form .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\int_{0}^{4} \frac{x}{x^{2}-1} d x=\frac{1}{2} \ln 15\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\int_{1}^{\infty} \frac{1}{x^{\sqrt{2}}} \ d x\) is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is convergent, then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The Midpoint Rule is always more accurate than the Trapezoidal Rule

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. (a) Every elementary function has an elementary derivative. (b) Every elementary function has an elementary antiderivative.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is continuous on and is convergent, then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If is a continuous, decreasing function on and , then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and are both convergent, then is convergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and are both divergent, then is divergent.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If and diverges, then also diverges.

Note: Additional practice in techniques of integration is provided in Exercises 7.5. Evaluate the integral. \(\int_{1}^{2} \frac{(x+1)^{2}}{x} d x\) Equation Transcription: Text Transcription: integral _1 ^2 (x+1)^2/x dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. Evaluate the integral. \(\int_{1}^{2} \frac{x}{(x+1)^{2}} d x\) Equation Transcription: Text Transcription: integral _1 ^2 x/(x+1)^2 dx

Note: Additional practice in techniques of integration is provided in Exercises 1-50 Evaluate the integral. \(\int \frac{e^{\sin x}}{\sec x} d x\) Equation Transcription: ?dx Text Transcription: integral e^sin x/sec x dx

Note: Additional practice in techniques of integration is provided in Exercises 1-50 Evaluate the integral. \(\int_{0}^{\pi / 6} t \sin 2 t \quad d t\) Equation Transcription: ?t sin 2t dt Text Transcription: integral _0 ^pi/6 t sin 2t dt

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\frac{x\left(x^{2}+4\right)}{x^{2}-4}\) can be put in the form \(\frac{A}{x+2}+\frac{B}{x-2}\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\frac{x^{2}+4}{x\left(x^{2}-4\right)}\) can be put in the form \(\frac{A}{x}+\frac{B}{x+2}+\frac{C}{x-2}\). Equation Transcription: Text Transcription: frac{x^2 + 4}{x(x^2 - 4)} frac{A}{x} + frac{B}{x + 2} + frac{C}{x - 2}

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\frac{x^{2}+4}{x^{2}(x-4)}\) can be put in the form \(\frac{A}{x^{2}}+\frac{B}{x-4}\). Equation Transcription: Text Transcription: frac{x^2 + 4}{x^2(x - 4)} frac{A}{x^2} + frac{B}{x - 4}

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\frac{x^{2}-4}{x\left(x^{2}+4\right)}\) can be put in the form \(\frac{A}{x}+\frac{B}{x^{2}+4}\). Equation Transcription: Text Transcription: frac{x^2} - 4}{xt(x^2 + 4)} frac{A}{x} + frac{B}{x^2 + 4}

A function \(f\) is defined by \(f(x)=\int_{0}^{\pi} \cos t \cos (x-t) d t, 0 \leqslant x \leqslant 2 \pi\). Find the minimum value of \(f\). ________________ Equation Transcription: 0 ? ? Text Transcription: f f(x) = int_{0}^\pi} cos t cos (x - t) dt, 0 leqslant x leqslant 2 pi f

If \(n\) is a positive integer, prove that \(\int_{0}^{1}(\ln x)^{n} d x=(-1)^{n} n !\). ________________ Equation Transcription: Text Transcription: n int_{0}^{1}(\ln x)^{n} d x=(-1)^{n} n !

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(\int_{-\infty}^{\infty} f(x) d x\) is convergent, then \(\int_{0}^{\infty} f(x) d x\) is convergent. Equation Transcription: Text Transcription: int_{-infty}^{infty} f(x) dx int_{0}^{infty} f(x) dx

Suppose that \(f\) is a positive function such that \(f^{\prime}\) is continuous. (a) How is the graph of \(y=f(x) \sin n x\) related to the graph of \(y=f(x)\) ? What happens as \(n \rightarrow \infty\) ? (b) Make a guess as to the value of the limit \(\lim _{n \rightarrow \infty} \int_{0}^{1} f(x) \sin n x d x\) based on graphs of the integrand. (c) Using integration by parts, confirm the guess that you made in part (b). [Use the fact that, since \(f^{\prime}\) is continuous, there is a constant \(M\) such that \(\left|f^{\prime}(x)\right| \leqslant M\) for \(0 \leqslant x \leqslant 1\).] ________________ Equation Transcription: ? for 0 ? ? 1 Text Transcription: f f’ y=f(x) \sin n x y=f(x) n rightarrow infty lim _{n rightarrow infty} \int_{0}^{1} f(x) sin nx dx f’ M |f^{\prime}(x)| leqslant M for 0 leqslant x leqslant 1

If \(0<a<b\), find \(\lim _{t \rightarrow 0}\left\{\int_{0}^{1}[b x+a(1-x)]^{t} d x\right\}^{1 / t}\) ________________ Equation Transcription: Text Transcription: 0 < a < b lim _{t rightarrow 0}{int_{0}^{1}[bx + a(1 - x)]^{t} dx}^{1/t}

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f\) is continuous on \([0, \infty)\) and \(\int_{1}^{\infty} f(x) d x\) is convergent, then \(\int_{0}^{x} f(x) d x\) is convergent. Equation Transcription: Text Transcription: f [0, infty) int_{1}^{infty} f(x) dx int_{0}^{x} f(x) dx

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f\) is a continuous, decreasing function on \([1, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=0\), then \(\int_{1}^{\infty} f(x) d x\) is convergent. Equation Transcription: Text Transcription: f [1, infty) lim _{x rightarrow infty} f(x) = 0 int_{1}^{infty} f(x) dx

1-50 Evaluate the integral. \(\int \frac{x^{2}+2}{x+2} \ d x\)

The circle with radius 1 shown in the figure touches the curve \(y = | 2x |\) twice. Find the area of the region that lies between the two curves. Equation Transcription: Text Transcription: y = | 2x |

A rocket is fired straight up, burning fuel at the constant rate of \(b\) kilograms per second. Let \(v=v(t)\) be the velocity of the rocket at time \(t\) and suppose that the velocity \(u\) of the exhaust gas is constant. Let \(M=M(t)\) be the mass of the rocket at time \(t\) and note that \(M\) decreases as the fuel burns. If we neglect air resistance, it follows from Newton's Second Law that \(F=M \frac{d v}{d t}-u b\) where the force \(F=-M g\). Thus Let \(M_{1}\) be the mass of the rocket without fuel, \(M_{2}\) the initial mass of the fuel, and \(M_{0}=M_{1}+M_{2}\). Then, until the fuel runs out at time \(t=M_{2} / b\), the mass is \(M=M_{0}-b t\). (a) Substitute \(M=M_{0}-b t\) into Equation 1 and solve the resulting equation for \(v\). Use the initial condition \(v(0)=0\) to evaluate the constant. (b) Determine the velocity of the rocket at time \(t=M_{2} / b\). This is called the burnout velocity. (c) Determine the height of the rocket \(y=y(t)\) at the burnout time. (d) Find the height of the rocket at any time \(t\). Equation Transcription: Text Transcription: M = M(t) v = v(t) F = ?Mt F=M dv/dt-ub M dv/dt-ub=-Mg

1-50 Evaluate the integral. \(\int \frac{x-1}{x^{2}+2 x} \ d x\)

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{\sec ^{6} \theta}{\tan ^{2} \theta} d \theta\) Equation Transcription: ?d Text Transcription: integral sec^6 theta/tan^2 theta d theta

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int x \cosh x \quad d x\) Equation Transcription: ? x cosh x dx Text Transcription: integral x cosh x dx

Note: Additional practice in techniques of integration is provided in Exercises 1-50 Evaluate the integral.

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{d x}{\sqrt{x^{2}-4 x}}\) ________________ Equation Transcription: Text Transcription: int frac{d x}{\sqrt{x^{2}-4 x}}

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{2^{\sqrt{x}}}{\sqrt{x}} d x\) ________________ Equation Transcription: Text Transcription: int frac{2^{sqrt{x}}}{\sqrt{x}} dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{x+1}{9 x^{2}+6 x+5} d x\) ________________ Equation Transcription: Text Transcription: int frac{x + 1}{9x^{2} + 6x + 5} dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \tan ^{5} \theta \sec ^{3} \theta \quad d \theta\) Equation Transcription: ? tan5 sec3 d Text Transcription: integral tan^5 theta sec^3 theta d theta

Note: Additional practice in techniques of integration is provided in Exercises 7.5. Evaluate the integral. \(\int_{0}^{2} \sqrt{x^{2}-2 x+2} d x\) Equation Transcription: Text Transcription: integral _0 ^2 square root x^2-2x+2 dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \cos \sqrt{t} \quad d t\) Equation Transcription: ? cos dt Text Transcription: integral cos sqrt t dt

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{d x}{x \sqrt{x^{2}+1}}\) ________________ Equation Transcription: Text Transcription: int frac{d x}{x sqrt{x^{2} + 1}}

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int e^{x} \cos x d x\)

1-50 Evaluate the integral. \(\int \frac{x \sin \left(\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} \ d x\)

1-50 Evaluate the integral. \(\int \frac{d x}{x^{1 / 2}+x^{1 / 4}}\)

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{3 x^{3}-x^{2}+6 x-4}{\left(x^{2}+1\right)\left(x^{2}+2\right)} d x\) ________________ Equation Transcription: Text Transcription: int frac{3 x^{3} - x^{2}+6 x-4}{(x^{2}+1)(x^{2}+2)} dx

1-50 Evaluate the integral. \(\int x \sin x \cos x \ d x\)

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int_{0}^{\pi / 2} \cos ^{3} x \sin 2 x \quad d x\) Equation Transcription: ? cos3 x sin 2x dx Text Transcription: integral _0 ^pi/2 cos^3 x sin 2x dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{\sqrt[3]{x}+1}{\sqrt[3]{x}-1} d x\) ________________ Equation Transcription: Text Transcription: int frac{sqrt[3]{x} + 1}{sqrt[3]{x} - 1} dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int_{-3}^{3} \frac{x}{1+|x|} d x\) ________________ Equation Transcription: Text Transcription: int_{-3}^{3} frac{x}{1 + |x|} dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{d x}{e^{x} \sqrt{1-e^{-2 x}}}\) ________________ Equation Transcription: Text Transcription: int frac{d x}{e^{x} sqrt{1-e^{-2 x}}}

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int_{0}^{\ln 10} \frac{e^{x} \sqrt{e^{x}-1}}{e^{x}+8} d x\) Equation Transcription: ?dx Text Transcription: integral _0 ^ln 10 e^x sqrt e^x - 1/e^x + 8 dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int_{0}^{\pi / 4} \frac{x \sin x}{\cos ^{3} x} d x\) Equation Transcription: ?dx Text Transcription: integral _0 ^pi/4 x sin x/cos^3 x dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{x^{2}}{\left(4-x^{2}\right)^{3 / 2}} d x\) ________________ Equation Transcription: Text Transcription: int frac{x^{2}}{(4-x^{2})^{3/2}} dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int(\arcsin x)^{2} \quad d x\) Equation Transcription: ? (arcsin x)2 dx Text Transcription: integral (arcsin x)^2 dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{1}{\sqrt{x+x^{3 / 2}}} d x\)

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{1-\tan \theta}{1+\tan \theta} d \theta\) Equation Transcription: ?d Text Transcription: integral 1-tan theta/1+tan theta d theta

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int(\cos x+\sin x)^{2} \cos 2 x \quad d x\) Equation Transcription: ? (cos x + sin x)2 cos 2x dx Text Transcription: integral (cos x + sin x)^2 cos 2x dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int x \cos ^{3}\left(x^{2}\right) \sqrt{\sin \left(x^{2}\right)} \quad d x\) Equation Transcription: ? x cos3(x2) dx Text Transcription: integral x cos^3 (x^2) sqrt sin(x^2) dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. Evaluate the integral. \(\int_{0}^{1 / 2} \frac{x e^{2 x}}{(1+2 x)^{2}} d x\) Equation Transcription: Text Transcription: integral _0 ^1/2 xe^2x/(1+2x)^2 dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int_{\pi / 4}^{\pi / 3} \frac{\sqrt{\tan \theta}}{\sin 2 \theta} d \theta\) Equation Transcription: ? d Text Transcription: integral _pi/4 ^pi/3 sqrt tan theta/sin 2 theta d theta

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int \frac{1}{\sqrt{e^{x}-4}} d x\) ________________ Equation Transcription: Text Transcription: int frac{1}{sqrt{e^{x} - 4}} dx

Note: Additional practice in techniques of integration is provided in Exercises 7.5. 1-50 Evaluate the integral. \(\int x \sin \left(\sqrt{1+x^{2}}\right) \quad d x\) Equation Transcription: ? x sin () dx Text Transcription: integral x sin(sqrt 1+x^2) dx

Evaluate the integral or show that it is divergent. \(\int_{1}^{\infty} \frac{1}{(2 x+1)^{3}} d x\) ________________ Equation Transcription: Text Transcription: int_{1}^{infty} frac{1}{(2x + 1)^{3}} dx

Evaluate the integral or show that it is divergent. \(\int_{1}^{\infty} \frac{\ln x}{x^{4}} d x\) Equation Transcription: ? dx Text Transcription: integral _1 ^infinity ln x/x^4 dx

Evaluate the integral or show that it is divergent. \(\int_{2}^{\infty} \frac{d x}{x \ln x}\) Equation Transcription: ? Text Transcription: integral _2 ^infinity dx/x ln x

Evaluate the integral or show that it is divergent. \(\int_{2}^{6} \frac{y}{\sqrt{y-2}} d y\) ________________ Equation Transcription: Text Transcription: int_{2}^{6} frac{y}{sqrt{y-2}} dy

Evaluate the integral or show that it is divergent. \(\int_{0}^{4} \frac{\ln x}{\sqrt{x}} d x\) Equation Transcription: ? dx Text Transcription: integral _0 ^4 ln x/sqrt x dx

Evaluate the integral or show that it is divergent. \(\int_{0}^{1} \frac{1}{2-3 x} d x\) Equation Transcription: Text Transcription: integral _0 ^1 ½--3x dx

Evaluate the integral or show that it is divergent. \(\int_{0}^{1} \frac{x-1}{\sqrt{x}} d x\)

Evaluate the integral or show that it is divergent. \(\int_{-1}^{\infty} \frac{d x}{x^{2}-2 x}\) Equation Transcription: Text Transcription: integral _-1 infinity dx/x^2-2x

Evaluate the integral or show that it is divergent. \(\int_{-\infty}^{\infty} \frac{d x}{4 x^{2}+4 x+5}\) ________________ Equation Transcription: Text Transcription: int_{-infty}^{infty} frac{dx}{4x^{2} + 4x + 5}

Evaluate the integral or show that it is divergent. \(\int_{1}^{\infty} \frac{\tan ^{-1} x}{x^{2}} d x\) ________________ Equation Transcription: Text Transcription: int_{1}^{infty} frac{tan ^{-1} x}{x^{2}} dx

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take \({C}=0\)). \(\int \ln \left(x^{2}+2 x+2\right) \quad d x\) Equation Transcription: C = 0 ? ln (x2 + 2x + 2) dx Text Transcription: C = 0 integral ln(x^2 + 2x +2) dx

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take \(C=0\) ). \(\int \frac{x^{3}}{\sqrt{x^{2}+1}} d x\) ________________ Equation Transcription: Text Transcription: C = 0 int frac{x^{3}}{sqrt{x^{2} + 1}} dx

Graph the function \(f(x)=\cos ^{2} x \sin ^{3} x\) and usc the graph to guess the value of the integral \(\int_{0}^{2 \pi} f(x) \quad d x\). Then evaluate the integral to confirm your guess. Equation Transcription: f(x) = cos2 x sin3 x ? f(x) dx Text Transcription: f(x) = cos^2 x sin^3 x integral _0 ^2pi f(x) dx

(a) How would you evaluate \(\int x^{5} e^{-2 x} d x\) by hand? (Don't actually carry out the integration.) (b) How would you evaluate \(\int x^{5} e^{-2 x} d x\) using a table of integrals? (Don't actually do it.) (c) Use a computer to evaluate \(\int x^{5} e^{-2 x} d x\). (d) Graph the integrand and the indefinite integral on the same screen ________________ Equation Transcription: Text Transcription: int x^{5} e^{-2 x} dx int x^{5} e^{-2 x} dx int x^{5} e^{-2 x} dx

Use the Table of Integrals on Reference Pages 6 - 10 to evaluate the integral. \(\int \sqrt{4 x^{2}-4 x-3} \ d x\)

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. \(\int \csc ^{5} t \quad d t\) Equation Transcription: ? csc5 t dt Text Transcription: integral csc^5 t dt

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. \(\int \cos x \sqrt{4+\sin ^{2} x} \quad d x\) Equation Transcription: ? cos x dx Text Transcription: integral cos x sqrt 4+sin^2 x dx

Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. \(\int \frac{\cot x}{\sqrt{1+2 \sin x}} \ d x\)

Verify Formula 33 in the Table of Integrals (a) by differentiation and (b) by using a trigonometric substitution.

Verify Formula 62 in the Table of Integrals.

Is it possible to find a number \(n\) such that \(\int_{0}^{\infty} x^{n} d x\) is convergent? ________________ Equation Transcription: Text Transcription: n int_{0}^{\infty} x^{n} dx

For what values of a is \(\int_{0}^{\infty} e^{a x} \cos x \ d x\) convergent? Evaluate the integral for those values of a.

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule with \(n = 10\) to approximate the given integral. Round your answers to six decimal places. \(\int_{2}^{4} \frac{1}{\ln x} d x\) ________________ Equation Transcription: Text Transcription: n = 10 int_{2}^{4} frac{1}{ln x} dx

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule with \(n = 10\) to approximate the given integral. Round your answers to six decimal places. \(\int_{1}^{4} \sqrt{x} \cos x d x\) ________________ Equation Transcription: Text Transcription: n = 10 int_{1}^{4} sqrt{x} cos x dx

Estimate the errors involved in Exercise 73, parts (a) and (b). How large should \(n\) be in each case to guarantee an error of less than 0.00001? ________________ Equation Transcription: Text Transcription: n = 10

Use Simpson's Rule with \(n=6\) to estimate the area under the curve \(y=e^{x} / x\) from \(x=1\) to \(x=4\). ________________ Equation Transcription: Text Transcription: n=6 y=e^{x} / x x=1 x=4

The speedometer reading (\(v\)) on a car was observed at 1-minute intervals and recorded in the chart. Use Simpson’s Rule to estimate the distance traveled by the car. ________________ Equation Transcription: Text Transcription: v

A population of honeybees increased at a rate of \(r(t)\) bees per week, where the graph of \(r\) is as shown. Use Simpson’s Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks. ________________ Equation Transcription: Text Transcription: r(t) r

(a) If \(f(x)=\sin (\sin x)\), use a computer algebra system to compute \(f^{(4)}(x)\) and then use a graph to find an upper bound for \(\left|f^{(4)}(x)\right|\). (b) Use Simpson's Rule with \(n=10\) to approximate \(\int_{0}^{\pi} f(x) d x\) and use part (a) to estimate the error. (c) How large should \(n\) be to guarantee that the size of the error in using \(S_{n}\) is less than 0.00001? ________________ Equation Transcription: Text Transcription: f(x)=\sin (\sin x) f^{(4)}(x) |f^{(4)}(x)| n = 10 int_{0}^{\pi} f(x) dx n S_{n}

Suppose you are asked to estimate the volume of a football. You measure and find that a football is 28 cm long. You use a piece of string and measure the circumference at its widest point to be 53 cm. The circumference 7 cm from each end is 45 cm. Use Simpson’s Rule to make your estimate.

Use the Comparison Theorem to determine whether the integral is convergent or divergent. (a) \(\int_{1}^{\infty} \frac{2+\sin x}{\sqrt{x}} d x\) (b) \(\int_{1}^{\infty} \frac{1}{\sqrt{1+x^{4}}} d x\) ________________ Equation Transcription: Text Transcription: int_{1}^{\infty} \frac{2+\sin x}{\sqrt{x}} dx int_{1}^{\infty} \frac{1}{\sqrt{1+x^{4}}} dx

Find the area of the region bounded by the hyperbola \(y^{2}-x^{2}=1\) and the line \(y=3\) ________________ Equation Transcription: Text Transcription: y^{2}-x^{2} = 1 y = 3

Find the area bounded by the curves \(y=\cos x\) and \(y=\cos ^{2} x\) between \(x=0\) and \(x=\pi\). ________________ Equation Transcription: Text Transcription: y = cos x y = cos ^{2} x x = 0 x = pi

Find the area of the region bounded by the curves \(y=1 /(2+\sqrt{x}), y=1 /(2-\sqrt{x})\), and \(x=1\) ________________ Equation Transcription: Text Transcription: y=1 /(2+\sqrt{x}), y=1 /(2-\sqrt{x}) x = 1

The region under the curve \(y=\cos ^{2} x, 0 \leqslant x \leqslant \pi / 2\), is rotated about the \(x\)-axis. Find the volume of the resulting solid. ________________ Equation Transcription: 0 ? ? Text Transcription: y= cos ^{2} x, 0 leqslant x leqslant pi/2 x

The region in Exercise 85 is rotated about the \(y\)-axis. Find the volume of the resulting solid. ________________ Equation Transcription: Text Transcription: y

If \(f^{\prime}\) is continuous on \([0, \infty)\) and \(\lim _{x \rightarrow \infty} f(x)=0\), show that \(\int_{0}^{\infty} f^{\prime}(x) d x=-f(0)\) ________________ Equation Transcription: Text Transcription: f’ [0, infty) lim _{x rightarrow infty} f(x)=0 int_{0}^{infty} f’(x) dx= -f(0)

We can extend our definition of average value of a continuous function to an infinite interval by defining the average value of \(f\) on the interval \([a, \infty)\) to be \(f_{\text {avg }}=\lim _{t \rightarrow \infty} \frac{1}{t-a} \int_{a}^{t} f(x) d x\) (a) Find the average value of \(y=\tan ^{-1} x\) on the interval \([0, \infty)\) (b) If \(f(x) \geqslant 0\) and \(\int_{a}^{\infty} f(x) d x\) is divergent, show that the average value of \(f\) on the interval \([a, \infty)\) is \(\lim _{x \rightarrow \infty} f(x)\) if this limit exists. (c) If \(\int_{a}^{\infty} f(x) d x\) is convergent, what is the average value of \(f\) on the interval \([a, \infty)\) ? (d) Find the average value of \(y=\sin x\) on the interval \([0, \infty)\). ________________ Equation Transcription: ? 0 Text Transcription: f [a, infty) f_ {avg }} = lim _{t rightarrow infty} frac{1}{t-a} int_{a}^{t} f(x) d x y=tan ^{-1} x [0, infty) f(x) geqslant 0 int_{a}^{infty} f(x) d x f [a, infty) lim _{x rightarrow infty} f(x) int_{a}^{infty} f(x) d x f [a, infty) y=sin x [0, infty)

Use the substitution \(u=1 / x\) to show that \(\int_{0}^{\infty} \frac{\ln x}{1+x^{2}} d x=0\) ________________ Equation Transcription: Text Transcription: u = 1/x int_{0}^{infty} frac{ln x}{1+x^{2}} dx = 0

The magnitude of the repulsive force between two point charges with the same sign, one of size 1 and the other of size \(q\), is \(F=\frac{q}{4 \pi \varepsilon_{0} r^{2}}\) where \(r\) is the distance between the charges and \(\varepsilon_{0}\) is a constant. The potential \(V\) at a point \(P\) due to the charge \(q\) is defined to be the work expended in bringing a unit charge to \(P\) from infinity along the straight line that joins \(q\) and \(P\). Find a formula for \(V\). ________________ Equation Transcription: Text Transcription: q F=frac{q}{4 \pi varepsilon_{0} r^{2}} r varepsilon_{0} V P q P q P V

Three mathematics students have ordered a 14-inch pizza. Instead of slicing it in the traditional way, they decide to slice it by parallel cuts, as shown in the figure. Being mathematics majors, they are able to determine where to slice so that each gets the same amount of pizza. Where are the cuts made?

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(\int x^{5} e^{x} d x\) can be evaluated by applying integration by parts five times.

Evaluate \(\int_{0}^{1}\left(\sqrt[3]{1-x^{7}}-\sqrt[7]{1-x^{3}}\right) \ d x\).

What is the form of the partial fraction decomposition of a rational function \(P(x) / Q(x)\) if the degree of \(P\) is less than the degree of \(Q\) and \(Q(x)\) has only distinct linear factors? What if a linear factor is repeated? What if \(Q(x)\) has an irreducible quadratic factor (not repeated)? What if the quadratic factor is repeated? Equation Transcription: Text Transcription: P(x) / Q(x) P Q Q(x) Q(x)

If f is an even function, r > 0, and a > 0, show that \(\int_{-r}^{r} \frac{f(x)}{1+a^{x}} \ d x=\int_{0}^{r} f(x) \ d x\) Hint: \(\frac{1}{1+u}+\frac{1}{1+u^{-1}}=1\)

The centers of two disks with radius 1 are one unit apart. Find the area of the union of the two disks.

An ellipse is cut out of a circle with radius a. The major axis of the ellipse coincides with a diameter of the circle and the minor axis has length 2b. Prove that the area of the remaining part of the circle is the same as the area of an ellipse with semiaxes a and a ? b.

A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L. The man keeps the rope straight and taut. The path followed by the boat is a curve called a tractrix and it has the property that the rope is always tangent to the curve (see the figure). (a) Show that if the path followed by the boat is the graph of the function y = f(x), then (b) Determine the function y = f(x).

A function f is defined by \(f(x)=\int_0^\pi \cos t \cos (x-t) d t, 0 \leqslant x \leqslant 2 \pi\). Find the minimum value of f.

If n is a positive integer, prove that \(\int_{0}^{1}(\ln x)^{n} d x=(-1)^{n} n !\)

Show that Hint: Start by showing that if In denotes the integral, then

Suppose that f is a positive function such that \(f^{\prime}\) is continuous. (a) How is the graph of \(y=f(x) \sin n x\) related to the graph of \(y=f(x)\) ? What happens as \(n \rightarrow \infty\) ? (b) Make a guess as to the value of the limit \(\lim _{n \rightarrow \infty} \int_{0}^{1} f(x) \sin n x \ d x\) based on graphs of the integrand. (c) Using integration by parts, confirm the guess that you made in part (b). [Use the fact that, since \(f^{\prime}\) is continuous, there is a constant M such that \(\left|f^{\prime}(x)\right| \leqslant M\) for \(0 \leqslant x \leqslant 1\).]

If 0 ? a ? b, find

Graph f(x) = sin(ex) and use the graph to estimate the value of t such that f(x)dx is a maximum. Then find the exact value of t that maximizes this integral.

Evaluate \(\int_{-1}^{\infty}\left(\frac{x^{4}}{1+x^{6}}\right)^{2} \ d x\).

Evaluate

The circle with radius 1 shown in the figure touches the curve y = | 2x | twice. Find the area of the region that lies between the two curves.

A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let v = v(t) be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let M = M(t) be the mass of the rocket at time t and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newton’s Second Law that where the force F = ?Mt. Thus Let M1 be the mass of the rocket without fuel, M2 the initial mass of the fuel, and M0 = M1 + M2. Then, until the fuel runs out at time t = M2/b, the mass is M = M0 ? bt. (a) Substitute M = M0 ? bt into Equation 1 and solve the resulting equation for v. Use the initial condition v(0) = 0 to evaluate the constant. (b) Determine the velocity of the rocket at time t = M2/b. This is called the burnout velocity. (c) Determine the height of the rocket y = y(t) at the burnout time. (d) Find the height of the rocket at any time t.