Chapter 29

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} \cos x d x\)

Evaluate the integrals. $$ \int x^{-2} \arctan x d x $$

Evaluate the integral. \(\int_{0}^{\frac{\pi}{4}} \sqrt{\tan x} \sec ^{2} x d x\)

Evaluate the integrals. $$ \int(\ln x)^{2} d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} x e^{-x} d x\)

Evaluate the integrals. $$ \int x^{2} \arctan x d x $$

Evaluate the integral. \(\int \tan x \sec ^{4} x d x\)

Find \(\int \cos ^{2} x d x\) in two ways. (a) Use the trigonometric identity \(\cos ^{2} x=\frac{1}{2}(1+\cos 2 x) .\) This is the most efficient way to do the problem. (b) Use integration by parts. You will solve algebraically for \(\int \cos ^{2} x d x\). (c) Check that your answers to parts (a) and (b) are correct by differentiating them. (d) Your answers to parts (a) and (b) are both antiderivatives of \(\cos ^{2} x\); therefore they must differ by a constant (where the constant is possibly zero). What is the constant?

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{-1}^{1} \frac{1}{5 x^{2}} d x\)

Evaluate the integrals. $$ \int \frac{x^{3}}{\sqrt{1-x^{2}}} d x $$

Evaluate the integral. \(\int \tan ^{3} x \sec ^{4} x d x\)

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int x^{5} \cos x^{3} d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{-\infty}^{\infty} \frac{1}{x^{3}} d x\)

Evaluate the integrals. $$ \int \ln \left(x^{2}-1\right) d x $$

Evaluate the integral. \(\int \tan ^{3} x \sec ^{5} x d x\)

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int(\ln x)^{3} d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{1} \ln x d x\)

Evaluate the integrals. $$ \int \frac{2 x^{3}-2 x^{2}+4 x+8}{(x-2)^{2}\left(x^{2}+3\right)} d x $$

Evaluate the integral. \(\int \tan ^{3} x \sec x d x\)

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int \sqrt{x} e^{\sqrt{x}} d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{\infty} \ln x d x\)

Evaluate the integrals. $$ \int \frac{x^{3}}{x^{2}+x-6} d x $$

Evaluate the integral. \(\int \frac{\tan ^{3} x}{\sec ^{4} x} d x\)

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int_{1}^{e} \frac{\ln x}{x} d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{\infty} \frac{1}{x(x+1)} d x\)

Evaluate the integral. \(\int \tan ^{8} x \sec ^{4} x d x\)

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int \frac{\ln x}{\sqrt{x}} d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} \frac{1}{x(x+1)} d x\)

Evaluate the integral. \(\int \frac{\sin x}{\cos ^{2} x} d x\)

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int_{0}^{\frac{\pi}{4}} \sec ^{2} x \tan x d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{e}^{\infty} \frac{1}{x \ln x} d x\)

Problems 22 through 24 refer to the information provided about Fourier series. \(A\) Fourier series expresses a function as a weighted infinite sum of terms of the form \(\sin n x\) and \(\cos n x\), where \(n\) is a nonnegative integer. Fourier series are a very powerful tool. In order to construct such a series we need the following results. \(m\) and \(n\) are positive integers. (a) \(\int_{-\pi}^{\pi} \sin m x \cos n x d x=0\) (b) \(\int_{-\pi}^{\pi} \sin m x \sin n x d x=\left\\{\begin{array}{ll}\pi & \text { if } m=n \\ 0 & \text { if } m \neq n\end{array}\right.\) (c) \(\int_{-\pi}^{\pi} \cos m x \cos n x d x=\left\\{\begin{array}{ll}\pi & \text { if } m=n \\ 0 & \text { if } m \neq n\end{array}\right.\) These results can be obtained using the product formulas given in this section. Prove statement (a) above.

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int_{1}^{e} \ln \sqrt{w} d w $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{e^{2}}^{\infty} \frac{1}{x(\ln x)^{2}} d x\)

Refer to the information provided about Fourier series. \(A\) Fourier series expresses a function as a weighted infinite sum of terms of the form \(\sin n x\) and \(\cos n x\), where \(n\) is a nonnegative integer. Fourier series are a very powerful tool. In order to construct such a series we need the following results. \(m\) and \(n\) are positive integers. (a) \(\int_{-\pi}^{\pi} \sin m x \cos n x d x=0\) (b) \(\int_{-\pi}^{\pi} \sin m x \sin n x d x=\left\\{\begin{array}{ll}\pi & \text { if } m=n \\ 0 & \text { if } m \neq n\end{array}\right.\) (c) \(\int_{-\pi}^{\pi} \cos m x \cos n x d x=\left\\{\begin{array}{ll}\pi & \text { if } m=n \\ 0 & \text { if } m \neq n\end{array}\right.\) These results can be obtained using the product formulas given in this section. Prove statement (b) above.

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int \sqrt{x} \ln x d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} \frac{x}{2+x^{2}} d x\)

Refer to the information provided about Fourier series. \(A\) Fourier series expresses a function as a weighted infinite sum of terms of the form \(\sin n x\) and \(\cos n x\), where \(n\) is a nonnegative integer. Fourier series are a very powerful tool. In order to construct such a series we need the following results. \(m\) and \(n\) are positive integers. (a) \(\int_{-\pi}^{\pi} \sin m x \cos n x d x=0\) (b) \(\int_{-\pi}^{\pi} \sin m x \sin n x d x=\left\\{\begin{array}{ll}\pi & \text { if } m=n \\ 0 & \text { if } m \neq n\end{array}\right.\) (c) \(\int_{-\pi}^{\pi} \cos m x \cos n x d x=\left\\{\begin{array}{ll}\pi & \text { if } m=n \\ 0 & \text { if } m \neq n\end{array}\right.\) These results can be obtained using the product formulas given in this section. Prove statement (c) above.

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int \sin (\ln x) d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} \frac{1}{\sqrt{x+1}} d x\)

Find the volume generated by revolving the region under one arch of \(\cos x\) about the \(x\) -axis.

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int_{0}^{1} \cos \sqrt{x} d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{-1}^{1} \frac{1}{\sqrt{x+1}} d x\)

Find the volume generated by revolving the region between the graphs of \(y=\sin x\) and \(y=\cos x\) from \(x=\frac{\pi}{4}\) to \(x=\frac{5 \pi}{4}\) around the horizontal line \(y=2\).

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int x^{3} \ln x d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{5} \frac{1}{x-3} d x\)

Find \(\int \frac{1}{\sin \theta} d \theta\).

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int_{0}^{\frac{\pi}{2}} x \sin x \cos x d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{\infty} \ln x d x\)

Evaluate \(\int \sec ^{3} x d x\) using a reduction formula.