Chapter 5

a. Multiply the numerator and denominator of sec \(x\) by \(\sec x+\tan x ;\) then use a change of variables to show that $$\int \sec x d x=\ln |\sec x+\tan x|+C$$ b. Use a change of variables to show that $$\int \csc x d x=-\ln |\csc x+\cot x|+C$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{8} \sqrt[3]{y} d y$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{\sqrt{2}}^{2} \frac{d x}{x \sqrt{x^{2}-1}}$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{1}^{2} \frac{z^{2}+4}{z} d z$$

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. $$\int\left(5 f^{3}(x)+7 f^{2}(x)+f(x)\right) f^{\prime}(x) d x$$

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\sqrt{3}} \frac{3 d x}{9+x^{2}}$$

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. $$\begin{aligned} &\int_{1}^{2}\left(5 f^{3}(x)+7 f^{2}(x)+f(x)\right) f^{\prime}(x) d x, \text { where } f(1)=4\\\ &f(2)=5 \end{aligned}$$

Find the area of the region \(R\) bounded by the graph of \(f\) and the \(x\) -axis on the given interval. Graph \(f\) and show the region \(R\) $$f(x)=2-|x| \text { on }[-2,4]$$

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. $$\int_{0}^{1} f^{\prime}(x) f^{\prime \prime}(x) d x, \text { where } f^{\prime}(0)=3 \text { and } f^{\prime}(1)=2$$

Find the area of the region \(R\) bounded by the graph of \(f\) and the \(x\) -axis on the given interval. Graph \(f\) and show the region \(R\) $$f(x)=\left(1-x^{2}\right)^{-1 / 2} \text { on }[-1 / 2, \sqrt{3} / 2]$$

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. \(\int\left(f^{(p)}(x)\right)^{n} f^{(p+1)}(x) d x,\) where \(p\) is a positive integer, \(n \neq-1\)

Find the area of the region \(R\) bounded by the graph of \(f\) and the \(x\) -axis on the given interval. Graph \(f\) and show the region \(R\) $$f(x)=x^{4}-4 \text { on }[1,4]$$

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. $$\int 2\left(f^{2}(x)+2 f(x)\right) f(x) f^{\prime}(x) d x$$

Occasionally, two different substitutions do the job. Use each substitution to evaluate the following integrals. $$\int_{0}^{1} x \sqrt{x+a} d x ; a>0 \quad(u=\sqrt{x+a} \text { and } u=x+a)$$

Simplify the given expressions. $$\int_{3}^{8} f^{\prime}(t) d t, \text { where } f^{\prime} \text { is continuous on }[3,8]$$

Occasionally, two different substitutions do the job. Use each substitution to evaluate the following integrals. $$\int_{0}^{1} x \sqrt[p]{x+a} d x ; a>0 \quad(u=\sqrt[p]{x+a} \text { and } u=x+a)$$

Simplify the given expressions. $$\frac{d}{d x} \int_{0}^{x^{2}} \frac{d t}{t^{2}+4}$$

Occasionally, two different substitutions do the job. Use each substitution to evaluate the following integrals. $$\int \sec ^{3} \theta \tan \theta d \theta \quad(u=\cos \theta \text { and } u=\sec \theta)$$

Simplify the given expressions. $$\frac{d}{d x} \int_{0}^{\cos x}\left(t^{4}+6\right) d t$$

\(\sin ^{2} a x\) and \(\cos ^{2} a x\) integrals Use the Substitution Rule to prove that $$\begin{aligned} &\int \sin ^{2} a x d x=\frac{x}{2}-\frac{\sin (2 a x)}{4 a}+C \text { and }\\\ &\int \cos ^{2} a x d x=\frac{x}{2}+\frac{\sin (2 a x)}{4 a}+C \end{aligned}$$

Simplify the given expressions. $$\frac{d}{d x} \int_{x}^{1} e^{t^{2}} d t$$

Integral of \(\sin ^{2} x \cos ^{2} x\) Consider the integral \(I=\int \sin ^{2} x \cos ^{2} x d x\) a. Find \(I\) using the identity \(\sin 2 x=2 \sin x \cos x\) b. Find \(I\) using the identity \(\cos ^{2} x=1-\sin ^{2} x\) c. Confirm that the results in parts (a) and (b) are consistent and compare the work involved in each method.

$$\frac{d}{d t}\left(\int_{1}^{t} \frac{3}{x} d x-\int_{t^{2}}^{1} \frac{3}{x} d x\right)$$

Substitution: shift Perhaps the simplest change of variables is the shift or translation given by \(u=x+c,\) where \(c\) is a real number. a. Prove that shifting a function does not change the net area under the curve, in the sense that $$\int_{a}^{b} f(x+c) d x=\int_{a+c}^{b+c} f(u) d u$$ b. Draw a picture to illustrate this change of variables in the case that \(f(x)=\sin x, a=0, b=\pi,\) and \(c=\pi / 2\)

$$\frac{d}{d t}\left(\int_{0}^{t} \frac{d x}{1+x^{2}}+\int_{0}^{1 / t} \frac{d x}{1+x^{2}}\right)$$

Substitution: scaling Another change of variables that can be interpreted geometrically is the scaling \(u=c x,\) where \(c\) is a real number. Prove and interpret the fact that $$\int_{a}^{b} f(c x) d x=\frac{1}{c} \int_{a c}^{b c} f(u) d u$$ Draw a picture to illustrate this change of variables in the case that \(f(x)=\sin x, a=0, b=\pi,\) and \(c=\frac{1}{2}\)

If necessary, use two or more substitutions to find the following integrals. \(\int x \sin ^{4} x^{2} \cos x^{2} d x\left(\text {Hint}: \text { Begin with } u=x^{2},\) then use \right. \(v=\sin u .)\)

Cubic zero net area Consider the graph of the cubic \(y=x(x-a)(x-b),\) where \(0

If necessary, use two or more substitutions to find the following integrals. \(\int \frac{d x}{\sqrt{1+\sqrt{1+x}}}(\text {Hint: Begin with } u=\sqrt{1+x} .)\)

Maximum net area What value of \(b>-1\) maximizes the integral $$\int_{-1}^{b} x^{2}(3-x) d x ?$$

If necessary, use two or more substitutions to find the following integrals. $$\int_{0}^{1} x \sqrt{1-\sqrt{x}} d x$$

Maximum net area Graph the function \(f(x)=8+2 x-x^{2}\) and determine the values of \(a\) and \(b\) that maximize the value of the integral $$\int_{a}^{b}\left(8+2 x-x^{2}\right) d x$$

If necessary, use two or more substitutions to find the following integrals. $$\int_{0}^{1} \sqrt{x-x \sqrt{x}} d x$$

An integral equation Use the Fundamental Theorem of Calculus, Part 1, to find the function \(f\) that satisfies the equation $$\int_{0}^{x} f(t) d t=2 \cos x+3 x-2$$ Verify the result by substitution into the equation.

If necessary, use two or more substitutions to find the following integrals. $$\int \tan ^{10} 4 x \sec ^{2} 4 x d x(\text { Hint: Begin with } u=4 x\text { .) }$$

Max / min of area functions Suppose \(f\) is continuous on \([0, \infty)\) and \(A(x)\) is the net area of the region bounded by the graph of \(f\) and the \(t\) -axis on \([0, x] .\) Show that the local maxima and minima of \(A\) occur at the zeros of \(f\). Verify this fact with the function \(f(x)=x^{2}-10 x\)

If necessary, use two or more substitutions to find the following integrals. $$\int_{0}^{\pi / 2} \frac{\cos \theta \sin \theta}{\sqrt{\cos ^{2} \theta+16}} d \theta(\text {Hint}: \text { Begin with } u=\cos \theta .)$$

Show that the sine integral \(S(x)=\int_{0}^{x} \frac{\sin t}{t} d t\) satisfies the (differential) equation \(x S^{\prime}(x)+2 S^{\prime \prime}(x)+x S^{\prime \prime \prime}(x)=0\)

Variable integration limits Evaluate \(\frac{d}{d x} \int_{-x}^{x}\left(t^{2}+t\right) d t\) (Hint: Separate the integral into two pieces.)

Discrete version of the Fundamental Theorem In this exercise, we work with a discrete problem and show why the relationship \(\int_{a}^{b} f^{\prime}(x) d x=f(b)-f(a)\) makes sense. Suppose we have a set of equally spaced grid points $$\left\\{a=x_{0}

Continuity at the endpoints Assume that \(f\) is continuous on \([a, b]\) and let \(A\) be the area function for \(f\) with left endpoint \(a .\) Let \(m^{*}\) and \(M^{*}\) be the absolute minimum and maximum values of \(f\) on \([a, b],\) respectively. a. Prove that \(m^{*}(x-a) \leq A(x) \leq M^{*}(x-a)\) for all \(x\) in \([a, b] .\) Use this result and the Squeeze Theorem to show that \(A\) is continuous from the right at \(x=a\) b. Prove that \(m^{*}(b-x) \leq A(b)-A(x) \leq M^{*}(b-x)\) for all \(x\) in \([a, b] .\) Use this result to show that \(A\) is continuous from the left at \(x=b\)