In Exercises 87 and 88, use a computer algebra system to find the integrals for 1, 2

In Exercises 1–6, identify u and dv for finding the integral using integration by parts. (Do not evaluate the integral.) \(\int x e^{2 x} d x\) Text Transcription: int xe^2x dx

In Exercises 1–6, identify u and dv for finding the integral using integration by parts. (Do not evaluate the integral.) \(\int x^{2} e^{2 x} d x\) Text Transcription: int x^2 e^2x dx

In Exercises 1–6, identify u and dv for finding the integral using integration by parts. (Do not evaluate the integral.) \(\int(\ln x)^{2} d x\) Text Transcription: int(ln x)^2 dx

In Exercises 1–6, identify u and dv for finding the integral using integration by parts. (Do not evaluate the integral.) \(\int \ln 5 x d x\) Text Transcription: int ln 5x dx

In Exercises 1–6, identify u and dv for finding the integral using integration by parts. (Do not evaluate the integral.) \(\int x \sec ^{2} x d x\) Text Transcription: int x sec^2 x dx

In Exercises 1–6, identify u and dv for finding the integral using integration by parts. (Do not evaluate the integral.) \(\int x^{2} \cos x d x\) Text Transcription: int x^2 cos x dx

In Exercises 7-10, evaluate the integral using integration by parts with the given choices of u and dv. \(\int x^{3} \ln x d x\); u=ln x, \(d v=x^{3} d x\) Text Transcription: int x^3 ln x dx dv=x^3 dx

In Exercises 7-10, evaluate the integral using integration by parts with the given choices of u and dv. \(\int(4 x+7) e^{x} d x\) ; u=4x+7, \(d v=e^{x} d x\) Text Transcription: int(4x+7) e^x dx dv=e^x dx

In Exercises 7-10, evaluate the integral using integration by parts with the given choices of u and dv. \(\int x \sin 3 x d x\) ; u=x, dv=sin 3x dx Text Transcription: int x sin 3x dx

In Exercises 7-10, evaluate the integral using integration by parts with the given choices of u and dv. \(\int x \cos 4 x d x\) ; u=x, dv=cos 4x dx Text Transcription: int x cos 4x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int x e^{-4 x} d x\) Text Transcription: int xe^-4x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \frac{4 x}{e^{x}} d x\) Text Transcription: int 4x/e^x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int x^{3} e^{x} d x\) Text Transcription: int x^3 e^x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \frac{e^{1 / t}}{t^{2}} d t\) Text Transcription: int e^1/t /t^2 dt

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int x^{2} e^{x^{3}} d x\) Text Transcription: int x^2 e^x^3 dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int x^{4} \ln x d x\) Text Transcription: int x^4 ln x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int t \ln (t+1) d t\) Text Transcription: int t ln (t+1) dt

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \frac{1}{x(\ln x)^{3}} d x\) Text Transcription: int 1/x(ln x)^3 dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \frac{(\ln x)^{2}}{x} d x\) Text Transcription: int (ln x)^2/x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \frac{\ln x}{x^{2}} d x\) Text Transcription: int ln x/x^2 dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \frac{x e^{2 x}}{(2 x+1)^{2}} d x\) Text Transcription: int xe^2x/(2x+1)^2 dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \frac{x^{3} e^{x^{2}}}{\left(x^{2}+1\right)^{2}} d x\) Text Transcription: int x^3 e^x^2/(x^2+1)^2 dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int\left(x^{2}-1\right) e^{x} d x\) Text Transcription: int(x^2-1) e^x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \frac{\ln 2 x}{x^{2}} d x\) Text Transcription: int ln 2x/x^2 dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int x \sqrt{x-5} d x\) Text Transcription: int x sqrt x-5 dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \frac{x}{\sqrt{5+4 x}} d x\) Text Transcription: int x/sqrt 5+4x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int x \cos x d x\) Text Transcription: int x cos x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int x \sin x d x\) Text Transcription: int x sin x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int x^{3} \sin x d x\) Text Transcription: int x^3 sin x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int x^{2} \cos x d x\) Text Transcription: int x^2 cos x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int t \csc t \cot t d t\) Text Transcription: int t csc t cot t dt

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \theta \sec \theta \tan \theta d \theta\) Text Transcription: int theta sec theta tan theta d theta

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int \arctan x d x\) Text Transcription: int arctan x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int 4 \arccos x d x\) Text Transcription: int 4 arccos x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int e^{2 x} \sin x d x\) Text Transcription: int e^2x sin x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int e^{-3 x} \sin 5 x d x\) Text Transcription: int e^-3x sin 5x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int e^{-x} \cos 2 x d x\) Text Transcription: int e^-x cos 2x dx

In Exercises 11-38, find the integral. (Note: Solve by the simplest method-not all require integration by parts.) \(\int e^{3 x} \cos 4 x d x\) Text Transcription: int e^3x cos 4x dx

In Exercises 39– 44, solve the differential equation. \(y^{\prime}=x e^{x^{2}}\) Text Transcription: y^prime=xe^x^2

In Exercises 39– 44, solve the differential equation. \(y^{\prime}=\ln x\) Text Transcription: y^prime=ln x

In Exercises 39– 44, solve the differential equation. \(\frac{d y}{d t}=\frac{t^{2}}{\sqrt{3+5 t}}\) Text Transcription: dy/dt=t^2/sqrt 3+5t

In Exercises 39– 44, solve the differential equation. \(\frac{d y}{d x}=x^{2} \sqrt{x-3}\) Text Transcription: dy/dx=x^2 sqrt x-3

In Exercises 39– 44, solve the differential equation. \((\cos y) y^{\prime}=2 x\) Text Transcription: (cos y) y^prime=2x

In Exercises 39– 44, solve the differential equation. \(y^{\prime}=\arctan \frac{x}{2}\) Text Transcription: y^prime=arctan x/2

Slope Fields In Exercises 45 and 46, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(\frac{d y}{d x}=x \sqrt{y} \cos x\), (0, 4) Text Transcription: dy/dx=x sqrt y cos x

Slope Fields In Exercises 45 and 46, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. \(\frac{d y}{d x}=e^{-x / 3} \sin 2 x\), \(\left(0, -\frac{18}{37}\right)\) Text Transcription: dy/dx=e^-x/3 sin 2x (0, -18/37)

Slope Fields In Exercises 47 and 48, use a computer algebra system to graph the slope field for the differential equation and graph the solution through the specified initial condition. \(\frac{d y}{d x}=\frac{x}{y} e^{x / 8}\) y(0)=2 Text Transcription: dy/dx=x/y e^x/8

Slope Fields In Exercises 47 and 48, use a computer algebra system to graph the slope field for the differential equation and graph the solution through the specified initial condition. \(\frac{d y}{d x}=\frac{x}{y} \sin x\) y(0)=4 Text Transcription: dy/dx=x/y sin x

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{3} x e^{x / 2} d x\) Text Transcription: int_0^3 xe^x/2 dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{2} x^{2} e^{-2 x} d x\) Text Transcription: int_0^2 x^2 e^-2x dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{\pi / 4} x \cos 2 x d x\) Text Transcription: int_0^pi/4 x cos 2x dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{\pi} x \sin 2 x d x\) Text Transcription: int_0^pi x sin 2x dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{1 / 2} \arccos x d x\) Text Transcription: int_0^1/2 arccos x dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{1} x \arcsin x^{2} d x\) Text Transcription: int_0^1 x arcsin x^2 dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{1} e^{x} \sin x d x\) Text Transcription: int_0^1 e^x sin x dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{2} e^{-x} \cos x d x\) Text Transcription: int_0^2 e^-x cos x dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{1}^{2} \sqrt{x} \ln x d x\) Text Transcription: int_1^2 sqrt x ln x dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{1} \ln \left(4+x^{2}\right) d x\) Text Transcription: int_0^1 ln (4+x^2) dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{2}^{4} x \operatorname{arcsec} x d x\) Text Transcription: int_2^4 x arcsec x dx

In Exercises 49-60, evaluate the definite integral. Use a graphing utility to confirm your result. \(\int_{0}^{\pi / 8} x \sec ^{2} 2 x d x\) Text Transcription: int_0^pi/8 x sec^2 2x dx

In Exercises 61– 66, use the tabular method to find the integral. \(\int x^{2} e^{2 x} d x\) Text Transcription: int x^2 e^2x dx

In Exercises 61– 66, use the tabular method to find the integral. \(\int x^{3} e^{-2 x} d x\) Text Transcription: int x^3 e^-2x dx

In Exercises 61– 66, use the tabular method to find the integral. \(\int x^{3} \sin x d x\) Text Transcription: int x^3 sin x dx

In Exercises 61– 66, use the tabular method to find the integral. \(\int x^{3} \cos 2 x d x\) Text Transcription: int x^3 cos 2x dx

In Exercises 61– 66, use the tabular method to find the integral. \(\int x \sec ^{2} x d x\) Text Transcription: int x sec^2 x dx

In Exercises 61– 66, use the tabular method to find the integral. \(\int x^{2}(x-2)^{3 / 2} d x\) Text Transcription: int x^2(x-2)^3/2 dx

In Exercises 67-74, find or evaluate the integral using substitution first, then using integration by parts. \(\int \sin \sqrt{x} d x\) Text Transcription: int sin sqrt x dx

In Exercises 67-74, find or evaluate the integral using substitution first, then using integration by parts. \(\int \cos \sqrt{x} d x\) Text Transcription: int cos sqrt x dx

In Exercises 67-74, find or evaluate the integral using substitution first, then using integration by parts. \(\int_{0}^{4} x \sqrt{4-x} d x\) Text Transcription: int_0^4 x sqrt 4-x dx

In Exercises 67-74, find or evaluate the integral using substitution first, then using integration by parts. \(\int 2 x^{3} \cos x^{2} d x\) Text Transcription: int 2x^3 cos x^2 dx

In Exercises 67-74, find or evaluate the integral using substitution first, then using integration by parts. \(\int x^{5} e^{x^{2}} d x\) Text Transcription: int x^5 e^x^2 dx

In Exercises 67-74, find or evaluate the integral using substitution first, then using integration by parts. \(\int_{0}^{2} e^{\sqrt{2 x}} d x\) Text Transcription: int_0^2 e^sqrt 2x dx

In Exercises 67-74, find or evaluate the integral using substitution first, then using integration by parts. \(\int \cos (\ln x) d x\) Text Transcription: int cos (ln x) dx

In Exercises 67-74, find or evaluate the integral using substitution first, then using integration by parts. \(\int \ln \left(x^{2}+1\right) d x\) Text Transcription: int ln(x^2+1) dx

WRITING ABOUT CONCEPTS Integration by parts is based on what differentiation rule? Explain.

WRITING ABOUT CONCEPTS In your own words, state how you determine which parts of the integrand should be u and dv.

WRITING ABOUT CONCEPTS When evaluating \(\int x \sin x d x\), explain how letting u=sin x and dv=x dx makes the solution more difficult to find. Text Transcription: int x sin x dx

CAPSTONE State whether you would use integration by parts to evaluate each integral. If so, identify what you would use for u and dv. Explain your reasoning. (a) \(\int \frac{\ln x}{x} d x\) (b) \(\int x \ln x d x\) (c) \(\int x^{2} e^{-3 x} d x\) (d) \(\int 2 x e^{x^{2}} d x\) (e) \(\int \frac{x}{\sqrt{x+1}} d x\) (f) \(\int \frac{x}{\sqrt{x^{2}+1}} d x\) Text Transcription: int ln x/x dx int x ln x dx int x^2 e^-3x dx int 2x e^x^2 dx int x/sqrt x+1 dx int x/sqrt x^2+1 dx

In Exercises 79-82, use a computer algebra system to (a) find or evaluate the integral and (b) graph two antiderivatives. (c) Describe the relationship between the graphs of the antiderivatives. \(\int t^{3} e^{-4 t} d t\) Text Transcription: int t^3 e^-4t dt

In Exercises 79-82, use a computer algebra system to (a) find or evaluate the integral and (b) graph two antiderivatives. (c) Describe the relationship between the graphs of the antiderivatives. \(\int \alpha^{4} \sin \pi \alpha d \alpha\) Text Transcription: int alpha^4 sin pi alpha d alpha

In Exercises 79-82, use a computer algebra system to (a) find or evaluate the integral and (b) graph two antiderivatives. (c) Describe the relationship between the graphs of the antiderivatives. \(\int_{0}^{\pi / 2} e^{-2 x} \sin 3 x d x\) Text Transcription: int_0^pi/2 e^-2x sin 3x dx

In Exercises 79-82, use a computer algebra system to (a) find or evaluate the integral and (b) graph two antiderivatives. (c) Describe the relationship between the graphs of the antiderivatives. \(\int_{0}^{5} x^{4}\left(25-x^{2}\right)^{3 / 2} d x\) Text Transcription: int_0^5 x^4(25-x^2)^3/2 dx

Integrate \(\int 2 x \sqrt{2 x-3} d x\) (a) by parts, letting \(d v=\sqrt{2 x-3} d x\). (b) by substitution, letting u=2x-3. Text Transcription: int 2x sqrt 2x-3 dx dv=sqrt 2x-3 dx

Integrate \(\int x \sqrt{9+x} d x\) (a) by parts, letting \(d v=\sqrt{9+x} d x\). b) by substitution, letting u=9+x. Text Transcription: int x sqrt 9+x dx dv=sqrt 9+x dx

Integrate \(\int \frac{x^{3}}{\sqrt{4+x^{2}}} d x\) (a) by parts, letting \(d v=\left(x / \sqrt{4+x^{2}}\right) d x\). (b) by substitution, letting \(u=4+x^{2}\). Text Transcription: int x^3/sqrt 4+x^2 dx dv=(x/sqrt 4+x^2) dx u=4+x^2

Integrate \(\int x \sqrt{4-x} d x\) (a) by parts, letting \(d v=\sqrt{4-x} d x\). (b) by substitution, letting u=4-x. Text Transcription: int x sqrt 4-x dx dv=sqrt 4-x dx

In Exercises 87 and 88, use a computer algebra system to find the integrals for n=0, 1, 2, and 3. Use the result to obtain a general rule for the integrals for any positive integer n and test your results for n=4. \(\int x^{n} \ln x d x\) Text Transcription: int x^n ln x dx

In Exercises 87 and 88, use a computer algebra system to find the integrals for n=0, 1, 2, and 3. Use the result to obtain a general rule for the integrals for any positive integer n and test your results for n=4. \(\int x^{n} e^{x} d x\) Text Transcription: int x^n e^x dx

In Exercises 89-94, use integration by parts to prove the formula. (For Exercises 89-92, assume that n is a positive integer.) \(\int x^{n} \sin x d x=-x^{n} \cos x+n \int x^{n-1} \cos x d x\) Text Transcription: int x^n sin x dx=-x^n cos x+n int x^n-1 cos x dx

In Exercises 89-94, use integration by parts to prove the formula. (For Exercises 89-92, assume that n is a positive integer.) \(\int x^{n} \cos x d x=x^{n} \sin x-n \int x^{n-1} \sin x d x\) Text Transcription: int x^n cos x dx=x^n sin x-n int x^n-1 sin x dx

In Exercises 89-94, use integration by parts to prove the formula. (For Exercises 89-92, assume that n is a positive integer.) \(\int x^{n} \ln x d x=\frac{x^{n+1}}{(n+1)^{2}}[-1+(n+1) \ln x]+C\) Text Transcription: int x^n ln x dx=x^n+1/(n+1)^2[-1+(n+1) ln x]+C

In Exercises 89-94, use integration by parts to prove the formula. (For Exercises 89-92, assume that n is a positive integer.) \(\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x\) Text Transcription: int x^n e^ax dx=x^n e^a x/a-n/a int x^n-1 e^ax dx

In Exercises 89-94, use integration by parts to prove the formula. (For Exercises 89-92, assume that n is a positive integer.) \(\int e^{a x} \sin b x d x=\frac{e^{a x}(a \sin b x-b \cos b x)}{a^{2}+b^{2}}+C\) Text Transcription: int e^ax sin bx dx=e^ax(a sin bx-b cos bx)a^2+b^2+C

In Exercises 89-94, use integration by parts to prove the formula. (For Exercises 89-92, assume that n is a positive integer.) \(\int e^{a x} \cos b x d x=\frac{e^{a x}(a \cos b x+b \sin b x)}{a^{2}+b^{2}}+C\) Text Transcription: int e^ax cos bx dx=e^ax(a cos bx+b sin bx/a^2+b^2+C

In Exercises 95-98, find the integral by using the appropriate formula from Exercises 89-94. \(\int x^{5} \ln x d x\) Text Transcription: int x^5 ln x dx

In Exercises 95-98, find the integral by using the appropriate formula from Exercises 89-94. \(\int x^{2} \cos x d x\) Text Transcription: int x^2 cos x dx

In Exercises 95-98, find the integral by using the appropriate formula from Exercises 89-94. \(\int e^{2 x} \cos 3 x d x\) Text Transcription: int e^2x cos 3x dx

In Exercises 95-98, find the integral by using the appropriate formula from Exercises 89-94. \(\int x^{3} e^{2 x} d x\) Text Transcription: int x^3 e^2x dx

Area In Exercises 99-102, use a graphing utility to graph the region bounded by the graphs of the equations, and find the area of the region. \(y=2 x e^{-x}\), y=0, x=3 Text Transcription: y=2xe^-x

Area In Exercises 99-102, use a graphing utility to graph the region bounded by the graphs of the equations, and find the area of the region. \(y=\frac{1}{16} x e^{-x / 4}\), y=0, x=0, x=4 Text Transcription: y=1/16 xe^-x/4

Area In Exercises 99-102, use a graphing utility to graph the region bounded by the graphs of the equations, and find the area of the region. \(y=e^{-x} \sin \pi x\), y=0, x=1 Text Transcription: y=e^-x sin pi x

Area In Exercises 99-102, use a graphing utility to graph the region bounded by the graphs of the equations, and find the area of the region. y=x sin x, y=0, \(x=\pi\) Text Transcription: x=pi

Area, Volume, and Centroid Given the region bounded by the graphs of y=ln x, y=0 and x=e find (a) the area of the region. (b) the volume of the solid generated by revolving the region about the x-axis. (c) the volume of the solid generated by revolving the region about the y-axis. (d) the centroid of the region.

Volume and Centroid Given the region bounded by the graphs of y=x sin x, y=0, x=0, and \(x=\pi\), find (a) the volume of the solid generated by revolving the region about the x-axis. (b) the volume of the solid generated by revolving the region about the y-axis. (c) the centroid of the region. Text Transcription: x=pi

Centroid Find the centroid of the region bounded by the graphs of y=arcsin x, x=0, and \(y=\pi / 2\). How is this problem related to Example 6 in this section? Text Transcription: y=pi/2

Centroid Find the centroid of the region bounded by the graphs of \(f(x)=x^{2}\), \(g(x)=2^{x}\), x=2, and x=4. Text Transcription: f(x)=x^2 g(x)=2^x

Average Displacement A damping force affects the vibration of a spring so that the displacement of the spring is given by \(y=e^{-4 t}(\cos 2 t+5 \sin 2 t)\). Find the average value of y on the interval from t=0 to \(t=\pi\). Text Transcription: y=e^-4t(cos 2t+5 sin 2t) t=pi

Memory Model A model for the ability M of a child to memorize, measured on a scale from 0 to 10 , is given by M=1+1.6 t ln t, \(0<t \leq 4\), where t is the child's age in years. Find the average value of this model (a) between the child's first and second birthdays. (b) between the child's third and fourth birthdays. Text Transcription: 0<t leq 4

Present Value In Exercises 109 and 110, find the present value P of a continuous income flow of c(t) dollars per year if \(P=\int_{0}^{t_{1}} c(t) e^{-r t} d t\) where \(t_{1}\) is the time in years and r is the annual interest rate compounded continuously. c(t)=100,000+4000 t, r=5%, \(t_{1}=10\) Text Transcription: P=int_0^t_1 c(t) e^-r t dt t_1 t_1=10

Present Value In Exercises 109 and 110, find the present value P of a continuous income flow of c(t) dollars per year if \(P=\int_{0}^{t_{1}} c(t) e^{-r t} d t\) where \(t_{1}\) is the time in years and r is the annual interest rate compounded continuously. c(t)=30,000+500 t, r=7%, \(t_{1}=5\) Text Transcription: P=int_0^t_1 c(t) e^-r t dt t_1 t_1=5

Integrals Used to Find Fourier Coefficients In Exercises 111 and 112 , verify the value of the definite integral, where n is a positive integer. \(\int_{-\pi}^{\pi} x \sin n x d x=\left\{\begin{array}{cc}\frac{2 \pi}{n}, & n \text { is odd } \\ -\frac{2 \pi}{n}, & n \text { is even }\end{array}\right\). Text Transcription: int_-pi^pi x sin nx dx=2 pi/n, -2 pi/n

Integrals Used to Find Fourier Coefficients In Exercises 111 and 112 , verify the value of the definite integral, where n is a positive integer. \(\int_{-\pi}^{\pi} x^{2} \cos n x d x=\frac{(-1)^{n} 4 \pi}{n^{2}}\) Text Transcription: int_-pi^pi x^2 cos nx dx=(-1)^n 4 pi/n^2

Vibrating String A string stretched between the two points (0,0) and (2,0) is plucked by displacing the string h units at its midpoint. The motion of the string is modeled by a Fourier Sine Series whose coefficients are given by \(b_{n}=h \int_{0}^{1} x \sin \frac{n \pi x}{2} d x+h \int_{1}^{2}(-x+2) \sin \frac{n \pi x}{2} d x\) . Find \(b_{n}\). Text Transcription: b_n=h int_0^1 x sin n pi x/2 dx+h int_1^2(-x+2) sin n pi x/2 dx b_n

Find the fallacy in the following argument that 0=1. \(\begin{aligned} &d v=d x \quad \Longrightarrow \quad v=\int d x=x \\ &u=\frac{1}{x} \quad \Rightarrow d u=-\frac{1}{x^{2}} d x \\ &0+\int \frac{d x}{x}=\left(\frac{1}{x}\right)(x)-\int\left(-\frac{1}{x^{2}}\right)(x) d x=1+\int \frac{d x}{x} \end{aligned}\) So, 0=1. Text Transcription: dv=dx Longrightarrow v=int dx=x u=1/x Rightarrow du=-1/x^2 dx 0+int dx/x=(1/x)(x)-int(-1/x^2)(x) dx=1+int dx/x

Let y=f(x) be positive and strictly increasing on the interval \(0<a \leq x \leq b\). Consider the region R bounded by the graphs of y=f(x), y=0, x=a, and x=b. If R is revolved about the y-axis, show that the disk method and shell method yield the same volume. Text Transcription: 0<a leq x leq b

Euler's Method Consider the differential equation \(f^{\prime}(x)=x e^{-x}\) with the initial condition f(0)=0. (a) Use integration to solve the differential equation. (b) Use a graphing utility to graph the solution of the differential equation. (c) Use Euler's Method with h=0.05, and the recursive capabilities of a graphing utility, to generate the first 80 points of the graph of the approximate solution. Use the graphing utility to plot the points. Compare the result with the graph in part (b). (d) Repeat part (c) using h=0.1 and generate the first 40 points. (e) Why is the result in part (c) a better approximation of the solution than the result in part (d)? Text Transcription: f^prime(x)=xe^-x

Euler's Method In Exercises 117 and 118, consider the differential equation and repeat parts (a)-(d) of Exercise 116. \(f^{\prime}(x)=3 x \sin (2 x)\) f(0)=0 Text Transcription: f^prime(x)=3x sin (2x)

Euler's Method In Exercises 117 and 118, consider the differential equation and repeat parts (a)-(d) of Exercise 116. \(f^{\prime}(x)=\cos \sqrt{x}\) f(0)=1 Text Transcription: f^prime(x)=cos sqrt x

Think About It Give a geometric explanation of why \(\int_{0}^{\pi / 2} x \sin x d x \leq \int_{0}^{\pi / 2} x d x\) Verify the inequality by evaluating the integrals. Text Transcription: int_0^pi/2 x sin x dx leq int_0^pi/2 x dx

Finding a Pattern Find the area bounded by the graphs of y=x sin x and y=0 over each interval. (a) \([0, \pi]\) (b) \([\pi, 2 \pi]\) (c) \([2 \pi, 3 \pi]\) Describe any patterns that you notice. What is the area between the graphs of y=x sin x and y=0 over the interval \([n \pi,(n+1) \pi]\), where n is any nonnegative integer? Explain. Text Transcription: [0, pi] [pi, 2 pi] [2 pi, 3 pi] [n pi, (n+1) pi]