Chapter 29

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. $$ \int_{0}^{2} e^{\sqrt{x}} d x $$

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. \(\int_{1}^{\infty} \frac{x}{\sqrt{3+x^{2}}} d x\)

In Problems 29 through 43, evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int x^{3} \sqrt{4-x^{2}} d x\)

(a) Show that \(\int \cos ^{n} x d x=\frac{1}{n} \cos ^{n-1} x \sin x+\frac{n-1}{n} \int \cos ^{n-2} x d x\), for \(n\) an integer, \(n \geq 2 .\) If you need some hints, see Exercise \(29.5 .\) (b) Use the results of part (a) to find \(\int \cos ^{2} x d x\). Verify your answer using differentiation. (c) Use the results of parts (a) and (b) to find \(\int \cos ^{6} x d x\).

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. \(\int_{1}^{\infty} \frac{\arctan x}{1+x^{2}} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{x^{3}}{\sqrt{9-x^{2}}} d x\)

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. \(\int_{1}^{2} \frac{1}{x \ln x} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{x}{\sqrt{4+x^{2}}} d x\)

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. \(\int_{1}^{\infty} \arctan x d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{x^{3}}{\sqrt{4+x^{2}}} d x\)

Show that \(\int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x\).

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. \(\int_{0}^{\pi} \tan x d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{1}{\sqrt{9+x^{2}}} d x\)

Show that \(\int x^{n} e^{x} d x=x^{n} e^{x}-n \int x^{n-1} e^{x} d x\).

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. \(\int_{0}^{\infty} \frac{x^{2}+3}{x+1} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int x^{2} \sqrt{x^{2}-9} d x\)

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. \(\int_{1}^{\infty} \frac{1}{(x+1)^{3}} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int_{0}^{\frac{3}{4}} \sqrt{9-4 x^{2}} d x\)

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. \(\int_{1}^{e^{2}} \frac{d x}{x \sqrt{\ln x}}\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \sqrt{4-9 x^{2}} d x\)

Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{1}^{\infty} \frac{(\sin x)^{2}}{x^{2}} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int x \sqrt{4-9 x^{2}} d x\)

Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{2}^{\infty} \frac{1}{x(x+1)} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int_{0}^{3} \frac{d x}{\left(4+x^{2}\right)^{\frac{3}{2}}}\)

Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{2}^{\infty} \frac{2}{x^{2} \ln x} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{\sqrt{x^{2}-4}}{x} d x\)

Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{1}^{\infty} \frac{1}{\sqrt{x^{7}+1}} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{x}{\sqrt{x^{2}-1}} d x\)

Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{0}^{\infty} \sin x e^{-x} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int_{\frac{2}{\sqrt{3}}}^{2} \frac{\sqrt{x^{2}-1}}{x} d x\)

Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{1}^{\infty} \frac{\cos x}{x^{2}} d x\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{2 x-3}{\sqrt{9-4 x^{2}}} d x\)

(a) Show that \(\int_{1}^{\infty} \frac{1}{1+x^{4}} d x\) converges. (b) Approximate \(\int_{1}^{\infty} \frac{1}{1+x^{4}} d x\) with error \(<0.01\). This involves making some choices, but the gist should be as follows. i. Snip off the tail, \(\int_{c}^{\infty} \frac{1}{1+x^{4}} d x\), for some constant \(c\). Bound it using \(\int_{c}^{\infty} \frac{1}{x^{4}} d x\). ii. Approximate \(\int_{1}^{c} \frac{1}{1+x^{4}} d x\) using numerical methods. iii. Be sure the sum of the bound in part (i) and the error in part (ii) is less than \(0.01 .\)

Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{x}{4-x^{2}} d x\)

The surface formed by revolving the graph of \(y=\frac{1}{x}\) on \([1, \infty)\) about the \(x\) -axis is known as Gabriel's horn. Find the volume of the horn. Curiously, you will find that the volume is finite even though the area under \(y=\frac{1}{x}\) on \([1, \infty)\) is infinite.

Find \(\int \sqrt{k^{2}-x^{2}} d x\) for any constant \(k\).

As mentioned at the beginning of this section, statisticians use probability density functions to determine the probability of a random variable falling in a certain interval. If \(p(x)\) is a probability density function, then \(p(x) \geq 0\) for all \(x\) and \(\int_{-\infty}^{\infty} p(x) d x=1 .\) A probability density function of the form \(p(x)=\left\\{\begin{array}{ll}\lambda e^{-\lambda x} & \text { for } x \geq 0, \\ 0 & \text { for } x<0\end{array}\right.\) where \(\lambda\) is a positive constant describes what is known as an exponential distribution. Verify that $$ \int_{-\infty}^{\infty} p(x) d x=1 $$

Show that the area enclosed by the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), where \(a\) and \(b\) are positive constants, is given by \(\pi a b\).

As mentioned at the beginning of this section, statisticians use probability density functions to determine the probability of a random variable falling in a certain interval. If \(p(x)\) is a probability density function, then \(p(x) \geq 0\) for all \(x\) and \(\int_{-\infty}^{\infty} p(x) d x=1 .\) A cumulative density function, \(C(x)\), gives the probability of a random variable taking on a value less than or equal to \(x .\) It is given by $$ C(x)=\int_{-\infty}^{x} p(y) d y $$ Show that for an exponential distribution (refer to Problem 45 ), the cumulative density function is given by $$ C(x)=\left\\{\begin{array}{ll} 1-e^{-\lambda x} & \text { for } x \geq 0 \\ 0 & \text { for } x<0 \end{array}\right. $$ Find \(\lim _{x \rightarrow \infty} C(x)\).

Find \(\int \sec ^{3} x d x\) as follows: Use integration by parts with \(u=\sec x .\) The resulting new integral will contain \(\tan ^{2} x .\) Replace \(\tan ^{2} x\) by \(\sec ^{2} x-1\) and split the integral into the difference of two integrals, \(\int \sec ^{3} x d x-\int \sec x d x .\) Integrate the latter and solve algebraically for the former.

Derive the reduction formula $$ \int \tan ^{n} x d x=\frac{\tan ^{n-1} x}{n-1}-\tan ^{n-2} x d x $$ where \(n\) is an integer greater than or equal to 2. (To do this, rewrite the integrand as \(\tan ^{n-2} x \cdot \tan ^{2} x\) and use a Pythagorean identity to convert \(\tan ^{2} x\) into an expression involving \(\sec x .)\)

As mentioned at the beginning of this section, statisticians use probability density functions to determine the probability of a random variable falling in a certain interval. If \(p(x)\) is a probability density function, then \(p(x) \geq 0\) for all \(x\) and \(\int_{-\infty}^{\infty} p(x) d x=1 .\) Suppose the number of minutes a caller spends on hold when calling a health clinic can be modeled using the probability density function. $$ p(x)=\left\\{\begin{array}{ll} 10 e^{-10 x} & \text { for } x \geq 0 \\ 0 & \text { for } x<0 \end{array}\right. $$ The probability that a random caller will wait at least 5 minutes on hold is given by \(\int_{5}^{\infty} p(x) d x .\) Find this probability. Note: it is not necessary to compute an improper integral in order to answer this question.

Derive the formula \(\sin A x \sin B x=\frac{1}{2}[\cos (A-B) x-\cos (A+B) x]\) given in this section. (Begin with the addition formula for cosine.)

Essay Question. Two of your classmates are having some trouble with improper integrals. Todd believes that improper integrals ought to diverge. He reasons that if \(f\) is positive, then the accumulated area keeps increasing, even if only by a little bit, so how can we get anything other than in nity? Dylan, on the other hand, is convinced that if \(\lim _{x \rightarrow \infty} f(x)=0\), then \(\int_{0}^{\infty} f(x) d x\) ought to converge. After all, he reasons, the rate at which area is accumulating is going to zero. Why isn \(\mathrm{t}\) that enough to assure convergence? Write an essay responding to Todd and Dylan s misconceptions. Your essay should be designed to help your classmates see the errors in their reasoning.