Chapter 5

In the following exercises, compute each indefinite integral. $$\int \frac{2}{x} d x$$

In the following exercises, compute each indefinite integral. $$\int \frac{1}{x^{2}} d x$$

In the following exercises, compute each indefinite integral. $$\int \frac{1}{\sqrt{x}} d x$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \frac{\ln x}{x} d x$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \frac{d x}{x(\ln x)^{2}}$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \frac{d x}{x \ln x}(x>1)$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \frac{d x}{x \ln x \ln (\ln x)}$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \tan \theta d \theta$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \frac{\cos x-x \sin x}{x \cos x} d x$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \frac{\ln (\sin x)}{\tan x} d x$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \ln (\cos x) \tan x d x$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int x e^{-x^{2}} d x$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int x^{2} e^{-x^{3}} d x$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int e^{\sin x} \cos x d x$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int e^{\tan x} \sec ^{2} x d x$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int e^{\ln x} \frac{d x}{x}$$

In the following exercises, find each indefinite integral by using appropriate substitutions. $$\int \frac{e^{\ln (1-t)}}{1-t} d t$$

In the following exercises, verify by differentiation that \(\int \ln x d x=x(\ln x-1)+C, \quad\) then use appropriate changes of variables to compute the integral. $$\int \ln x d x \text { (Hint: } \int \ln x d x=\frac{1}{2} \int x \ln \left(x^{2}\right) d x )$$

In the following exercises, verify by differentiation that \(\int \ln x d x=x(\ln x-1)+C, \quad\) then use appropriate changes of variables to compute the integral. \(\int x^{2} \ln ^{2} x d x\)

In the following exercises, verify by differentiation that \(\int \ln x d x=x(\ln x-1)+C, \quad\) then use appropriate changes of variables to compute the integral. $$\int \frac{\ln x}{x^{2}} d x \quad\left(\text {Hint} : \text { Set } u=\frac{1}{x} .\right)$$

In the following exercises, verify by differentiation that \(\int \ln x d x=x(\ln x-1)+C, \quad\) then use appropriate changes of variables to compute the integral. $$\int \frac{\ln x}{\sqrt{x}} d x \quad(\text {Hint} : \operatorname{Set} u=\sqrt{x} .)$$

In the following exercises, verify by differentiation that \(\int \ln x d x=x(\ln x-1)+C, \quad\) then use appropriate changes of variables to compute the integral. Write an integral to express the area under the graph of \(y=\frac{1}{t}\) from \(t=1\) to \(e^{x}\) and evaluate the integral.

Write an integral to express the area under the graph of \(y=\frac{1}{t}\) from \(t=1\) to \(e^{x}\) and evaluate the integral.

Write an integral to express the area under the graph of \(y=e^{t}\) between \(t=0\) and \(t=\ln x,\) and evaluate the integral.

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$\int \tan (2 x) d x$$

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$\int \frac{\sin (3 x)-\cos (3 x)}{\sin (3 x)+\cos (3 x)} d x$$

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$\int \frac{x \sin \left(x^{2}\right)}{\cos \left(x^{2}\right)} d x$$

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$\int x \csc \left(x^{2}\right) d x$$

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$\int \ln (\cos x) \tan x d x$$

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$\int \ln (\csc x) \cot x d x$$

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$\int \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}} d x$$

In the following exercises, evaluate the definite integral. $$\int_{1}^{2} \frac{1+2 x+x^{2}}{3 x+3 x^{2}+x^{3}} d x$$

In the following exercises, evaluate the definite integral. $$\int_{0}^{\pi / 4} \tan x d x$$

In the following exercises, evaluate the definite integral. $$\int_{0}^{\pi / 3} \frac{\sin x-\cos x}{\sin x+\cos x} d x$$

In the following exercises, evaluate the definite integral. $$\int_{\pi / 6}^{\pi / 2} \csc x d x$$

In the following exercises, evaluate the definite integral. $$\int_{\pi / 4}^{\pi / 3} \cot x d x$$

In the following exercises, integrate using the indicated substitution. $$\int \frac{x}{x-100} d x ; u=x-100$$

In the following exercises, integrate using the indicated substitution. $$\int \frac{y-1}{y+1} d y ; u=y+1$$

In the following exercises, integrate using the indicated substitution. $$\int \frac{1-x^{2}}{3 x-x^{3}} d x ; u=3 x-x^{3}$$

In the following exercises, integrate using the indicated substitution. $$\int \frac{\sin x+\cos x}{\sin x-\cos x} d x ; u=\sin x-\cos x$$

In the following exercises, integrate using the indicated substitution. \(\int e^{2 x} \sqrt{1-e^{2 x}} d x ; u=e^{2 x}\)

In the following exercises, integrate using the indicated substitution. $$\int \ln (x) \frac{\sqrt{1-(\ln x)^{2}}}{x} d x ; u=\ln x$$

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate \(R_{50}\) and solve for the exact area. $$y=e^{x} \text { over }[0,1]$$

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate \(R_{50}\) and solve for the exact area. [T] \(y=e^{-x}\) over [0,1]

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate \(R_{50}\) and solve for the exact area. $$y=\ln (x) \text { over }[1,2]$$

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate \(R_{50}\) and solve for the exact area. $$y=\frac{x+1}{x^{2}+2 x+6} \text { over }[0,1]$$

In the following exercises, does the right-endpoint approximation overestimate or underestimate the exact area? Calculate the right endpoint estimate \(R_{50}\) and solve for the exact area. $$y=2^{x} \text { over }[-1,0]$$

In the following exercises, \(f(x) \geq 0\) for \(a \leq x \leq b\) . Find the area under the graph of \(f(x)\) between the given values \(a\) and \(b\) by integrating. $$f(x)=\frac{\log _{10}(x)}{x} ; a=10, b=100$$

In the following exercises, \(f(x) \geq 0\) for \(a \leq x \leq b\) . Find the area under the graph of \(f(x)\) between the given values \(a\) and \(b\) by integrating. $$f(x)=\frac{\log _{2}(x)}{x} ; a=32, b=64$$

In the following exercises, \(f(x) \geq 0\) for \(a \leq x \leq b\) . Find the area under the graph of \(f(x)\) between the given values \(a\) and \(b\) by integrating. $$f(x)=2^{-x} ; a=1, b=2$$