Looking ahead (to Chapter 9) Suppose that a function f has

On which derivative rule is integration by parts based?

How would you choose the term dv when evaluating \(\int x^{n} e^{a x} d x\) using integration by parts

How would you choose the term u when evaluating \(\int x^{n} \cos a x d x\) using integration by parts?

Explain how integration by parts is used to evaluate a definite integral.

For what type of integrand is integration by parts useful?

How would you choose u and dv to simplify \(\int x^{4} e^{-2 x} d x\)?

Integration by parts Evaluate the following integrals \(\int x \cos x d x\)

Integration by parts Evaluate the following integrals \(\int x \sin 2 x d x\)

Integration by parts Evaluate the following integrals \(\int t e^{t} d t\)

Integration by parts Evaluate the following integrals \(\int 2 x e^{3 x} d x\)

Integration by parts Evaluate the following integrals \(\int x^{2} \sin 2 x d x\)

Integration by parts Evaluate the following integrals \(\int s e^{-2 s} d s\)

Integration by parts Evaluate the following integrals \(\int x^{2} e^{4 x} d x\)

Integration by parts Evaluate the following integrals \(\int \theta \sec ^{2} \theta d \theta\)

Integration by parts Evaluate the following integrals \(\int x^{2} \ln x d x\)

Integration by parts Evaluate the following integrals \(\int x \ln x d x\)

Integration by parts Evaluate the following integrals \(\int \frac{\ln x}{x^{10}} d x\)

Integration by parts Evaluate the following integrals \(\int \sin ^{-1} x d x\)

Integration by parts Evaluate the following integrals \(\int \tan ^{-1} x d x\)

Integration by parts Evaluate the following integrals \(\int x \sec ^{-1} x d x, x \geq 1\)

Integration by parts Evaluate the following integrals \(\int x \sin x \cos x d x\)

Integration by parts Evaluate the following integrals \(\int x \tan ^{-1}\left(x^{2}\right) d x\)

Repeated integration by parts Evaluate the following integrals \(\int e^{x} \cos x d x\)

Repeated integration by parts Evaluate the following integrals \(\int e^{3 x} \cos 2 x d x\)

Repeated integration by parts Evaluate the following integrals \(\int e^{-x} \sin 4 x d x\)

Repeated integration by parts Evaluate the following integrals \(\int x^{2} \ln ^{2} x d x\)

Repeated integration by parts Evaluate the following integrals \(\int t^{3} e^{-t} d t\)

Repeated integration by parts Evaluate the following integrals \(\int e^{-2 \theta} \sin 6 \theta d \theta\)

Repeated integration by parts Evaluate the following integral.

Definite integrals Evaluate the following definite integrals. \(\int_{1}^{e} \ln 2 x d x\)

Definite integrals Evaluate the following definite integrals. \(\int_{0}^{\pi / 2} x \cos 2 x d x\)

Definite integrals Evaluate the following definite integrals. \(\int_{0}^{\ln 2} x e^{x} d x\)

Definite integrals Evaluate the following definite integrals. \(\int_{1}^{e^{2}} x^{2} \ln x d x\)

Definite integrals Evaluate the following definite integrals. \(\int_{0}^{1 / \sqrt{2}} y \tan ^{-1} y^{2} d y\)

Definite integrals Evaluate the following definite integrals. \(\int_{1 / 2}^{\sqrt{3} / 2} \sin ^{-1} y d y\)

Definite integrals Evaluate the following definite integrals. \(\int_{2 / \sqrt{3}}^{2} z \sec ^{-1} z d z\)

Definite integrals Evaluate the following definite integral.

Volumes of solids Find the volume of the solid that is generated when the region is revolved as described. The region bounded by f(x) = sin x and the x-axis on [0,\(\pi\)] is revolved about the y-axis

Volumes of solids Find the volume of the solid that is generated when the region is revolved as described. The region bounded by f(x)= x ln x and the x-axis on [1,\(e^{2}\)] is revolved about the x-axis

Volumes of solids Find the volume of the solid that is generated when the region is revolved as described. The region bounded by \(f(x)=e^{-x}\) and the x-axis on [0,ln 2] is revolved about the line x = ln 2.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. \(\int u v^{\prime} d x=\left(\int u d x\right)\left(\int v^{\prime} d x\right)\) b. \(\int u v^{\prime} d x=u v-\int v u^{\prime} d x\)

Reduction formulas Use integration by parts to derive the following reduction formulas. \(\int x^{n} e^{a x} d x=\frac{x^{n} e^{a x}}{a}-\frac{n}{a} \int x^{n-1} e^{a x} d x \quad \text { for } a \neq 0\)

Reduction formulas Use integration by parts to derive the following reduction formulas. \(\int x^{n} \cos a x d x=\frac{x^{n} \sin a x}{a}-\frac{n}{a} \int x^{n-1} \sin a x d x \text { for } a \neq 0\)

Reduction formulas Use integration by parts to derive the following reduction formulas. \(\int x^{n} \sin a x d x=-\frac{x^{n} \cos a x}{a}+\frac{n}{a} \int x^{n-1} \cos a x d x \quad \text { for } a \neq 0\)

Reduction formulas Use integration by parts to derive the following reduction formulas. \(\int \ln ^{n} x d x=x \ln ^{n} x-n \int \ln ^{n-1} x d x\)

Applying reduction formulas Use the reduction formulas in Exercises 42-45 to evaluate the following integrals \(\int x^{2} e^{3 x} d x\)

Applying reduction formulas Use the reduction formulas in Exercises 42-45 to evaluate the following integrals \(\int x^{2} \cos 5 x d x\)

Applying reduction formulas Use the reduction formulas in Exercises 42-45 to evaluate the following integrals \(\int x^{3} \sin x d x\)

Applying reduction formulas Use the reduction formulas in Exercises 42-45 to evaluate the following integrals \(\int \ln ^{4} x d x\)

Integrals involving \(\int \ln x d x\) Use a substitution to reduce the following integrals to \(\int \ln u d u\). Then, evaluate the resulting integral. \(\int \cos x \ln (\sin x) d x\)

Integrals involving \(\int \ln x d x\) Use a substitution to reduce the following integrals to \(\int \ln u d u\). Then, evaluate the resulting integral. \(\int \sec ^{2} x \ln (\tan x+2) d x\)

Two methods a. Evaluate \(\int x \ln x^{2} d x\) using the substitution \(u = x^{2}\) and evaluating \(\int \ln u d u .\). b. Evaluate \(\int x \ln x^{2} d x\)using integration by parts. c. Verify that your answers to parts (a) and (b) are consistent.

Logarithm base b Prove that \(\int \log _{b} x d x=\frac{1}{\ln b}(x \ln x-x)+C\)

Two integration methods Evaluate \(\int \sin x \cos x d x\) using integration by parts. Then evaluate the integral using a substitution. Reconcile your answers.

Combining two integration methods Evaluate \(\int \cos (\sqrt{x})\) using a substitution followed by integration by parts.

Combining two integration methods Evaluate \(\int_{0}^{\pi^{2} / 4} \sin (\sqrt{x}) d x\) using a substitution followed by integration by parts.

Function defined as an integral Find the arc length of the function \(f(x)=\int_{e}^{x} \sqrt{\ln ^{2} t-1} d t \text { on }\left[e, e^{3}\right]\)

A family of exponentials The curves \(y=x e^{-a x}\) are shown in the figure for a = 1, 2, and 3. a. Find the area of the region bounded by \(y=x e^{-x}\) and the x-axis on the interval (0,4). b. Find the area of the region bounded by \(y=x e^{-a x}\) and the x-axis on the interval (0,4), where a > 0. c. Find the area of the region bounded by \(y=x e^{-a x}\) and the X-axis on the interval [0,b]. Because this area depends on a and b, we call it A(a,b), where a > 0 and b > 0. d. Use part (c) to show that A(1, In b) = 4A(2, (In b)/2). e. Does this pattern continue? Is it true that \(A(1, \ln b)=a^{2} A(a,(\ln b) / a)\)?

Solid of revolution Find the volume of the solid generated when the region bounded by y = cos x and the x-axis on the interval [0, \(\pi\)/2] is revolved about the y-axis.

Between the sine and inverse sine Find the area of the region bounded by the curves y = sin x and \(y=\sin ^{-1} x\) on the interval [0,1]

Comparing volumes Let R be the region bounded by y = sinx and the x-axis on the interval (0,\(\pi\)). Which is greater, the volume of the solid generated when R is revolved about the x-axis or the volume of the solid generated when R is revolved about the y-axis?

Log integrals Use integration by parts to show that for \(m \neq -1\), \(\int x^{m} \ln x d x=\frac{x^{m+1}}{m+1}\left(\ln x-\frac{1}{m+1}\right)+C\) And for m = -1 \(\int \frac{\ln x}{x} d x=\frac{1}{2} \ln ^{2} x+C\)

A useful integral a. Use integration by parts to show that if f’ is continuous \(\int x f^{\prime}(x) d x=x f(x)-\int f(x) d x\) b. Use part (a) to evaluate \(\int x e^{3 x} d x\)

Integrating inverse functions a. Let \(y = f^{-1}(x)\), which means x = f(y) and dx = f’(y)dy. Show that \(\int f^{-1}(x) d x=\int y f^{\prime}(y) d y\) b. Use the result of Exercise 63 top show that \(\int f^{-1}(x) d x=y f(y)-\int f(y) d y\) c. Use the result of part (b) to evaluate \(\int \ln x d x\) (express the result in terms of x) d. Use the result of part (b) to evaluate \(\int \sin ^{-1} x d x\) e. Use the result of part (b) to evaluate \(\int \tan ^{-1} x d x\)

Integral of \(\sec ^{3} x\) Use integration by parts to show that \9\int \sec ^{3} x d x=\frac{1}{2} \sec x \tan x+\frac{1}{2} \int \sec x d x\)

Two useful exponential integrals Use integration by parts to derive the following formulas for real numbers a and b. \(\int e^{a x} \sin b x d x=\frac{e^{a x}(a \sin b x-b \cos b x)}{a^{2}+b^{2}}+C \) \(\int e^{a x} \cos b x d x=\frac{e^{a x}(a \cos b x+b \sin b x)}{a^{2}+b^{2}}+C\)

Oscillator displacements Suppose an oscillator (such as a pendulum or a mass on a spring) that is slowed by friction has the position function \(s(t)=e^{-t} \sin t\). a. Graph the position function. At what times does the oscillator pass through the position s = 0? b. Find the average value of the position on the interval [0, \(\pi\)]. c. Generalize part (b) and find the average value of the position on the interval \([n \pi,(n+1) \pi] \text { for } n=0,1,2, \ldots\) d. Let \(a_{n}\) be the absolute value of the average position on the intervals \([n \pi,(n+1) \pi] \text { for } n=0,1,2, \ldots\) Describe the pattern in the numbers \(a_{0}, a_{1}, a_{2}, \ldots).

Find the error Suppose you evaluate \(\int \frac{d x}{x}\) using integration by parts. With u=1/x and dv=dx, you find that \(du = -1/x^{2} dx\), v=x, and \(\int \frac{d x}{x}=\left(\frac{1}{x}\right) x-\int x\left(-\frac{1}{x^{2}}\right) d x=1+\int \frac{d x}{x}\) You conclude that 0=1. Explain the problem with the calculation.

Proof without words Explain how the diagram in the figure illustrates integration by parts for definite integrals.

An identity Show that if f and g have continuous second derivatives and f(0) = f(1) = g(0) = g(1) = 0, then \(\int_{0}^{1} f^{\prime \prime}(x) g(x) d x=\int_{0}^{1} f(x) g^{\prime \prime}(x) d x)

Possible and impossible integrals Let \(I_{n}=\int x^{n} e^{-x^{2}} d x\), where n is a nonnegative integer. a. \(I_{0}=\int e^{-x^{2}} d x) cannot be expressed in terms of elementary functions. Evaluate \(I_{1}\). b. Use integration by parts to evaluate \(I_{3}). c. Use integration by parts and the result of part (b) to evaluate \(I_{5}\). d. Show that, in general, if n is odd, then \(I_{n}=-\frac{1}{2} e^{-x^{2}} p_{n-1}(x)\), where \(p_{n-1}) is an even polynomial of degree n - 1. e. Argue that if n is even, then \(I_{n}\) cannot be expressed in terms of elementary functions.

Looking ahead (to Chapter 9) Suppose that a function has derivatives of all orders near r = 0. By the Fundamental Theorem of Calculus, \(f(x)-f(0)=\int_{0}^{x} f^{\prime}(t) d t\) a. Evaluate the integral using integration by parts to show that \(f(x)=f(0)+x f^{\prime}(0)+\int_{0}^{x} f^{\prime \prime}(t)(x-t) d t\) b. Show (by observing a pattern or using induction) that by inte grating by parts n times, \(f(x)=f(0)+x f^{\prime}(0)+\frac{1}{2 !} x^{2} f^{\prime \prime}(0)+\cdots+\frac{1}{n !} x^{n} f^{(n)}(0)+\frac{1}{n !} \int_{0}^{x} f^{(n+1)}(t)(x-t)^{n} d t+\cdots\) This expression, called the Taylor series for f at x = 0, is revisited in Chapter 9.