Chapter 7

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{x-2}{(x-1)^{2}} d x $$

In Problems \(43-58\), use substitution to evaluate each definite integral. $$ \int_{0}^{3} x \sqrt{x^{2}+1} d x $$

Find a comparison function for each integrand and determine whether the integral is convergent. $$ \int_{0}^{\infty} \frac{1}{e^{x}+e^{-x}} d x $$

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int \sin x \cos x e^{\sin x} d x $$

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{2 x^{2}-2 x+1}{x^{2}(x-1)} d x $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{2} x^{5} \sqrt{x^{3}+2} d x $$

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int_{0}^{\pi / 4} \cos \sqrt{x} d x $$

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{2 x^{2}+x+1}{x(x+1)^{2}} d x $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{1}^{3} \frac{4 x+6}{\left(x^{2}+3 x\right)^{3}} d x $$

(a) Show that $$\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0$$ (b) Use your result in (a) to show that, for any \(c>0\), $$c x \geq \ln x$$ for sufficiently large \(x\).46. (a) Show that $$\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0$$ (b) Use your result in (a) to show that, for any \(c>0\), $$c x \geq \ln x$$ for sufficiently large \(x\). (c) Use your result in (b) to show that, for any \(p>0\), $$x^{p} e^{-x} \leq e^{-x / 2}$$ provided that \(x\) is sufficiently large. (d) Use your result in (c) to show that, for any \(p>0\), $$\int_{0}^{\infty} x^{p} e^{-x} d x$$ is convergent.

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int_{0}^{\pi^{2}} \sin \sqrt{x} d x $$

Use partial fraction decompositions to evaluate each integral. $$ \int \frac{2 x^{3}-x-1}{x^{2}(x+1)^{2}} d x $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{2} \frac{2 x}{\left(4 x^{2}+2\right)^{1 / 3}} d x $$

Schwabe and Bruggeman (2014) modeled how yeast cells respond to a change in the amount of nutrient available in their environment. Schwabe and Bruggeman found that the time taken by the yeast cells to respond to an increase in the amount of nutrient available in their environment could be modeled by a Gamma distributed random variable. Specifically the probability that a cell responds in time \(t\) is proportional to \(p(t)=t^{a-1} e^{-b t}\), where \(a\) and \(b\) are both positive constants. It can be shown (see Chapter 12) that the probability a cell responds at all (i.e., in finite time) to the change in environmental conditions is proportional to $$\int_{0}^{\infty} p(t) d t$$ (a) Assume \(a=1\); show that the integral \(\int_{0}^{\infty} p(t) d t\) is convergent and find its value. (b) Now assume \(a=2\); again show that the integral is convergent, and find its value. (c) If \(a=3 / 2\), you cannot use integration by parts to find the value of the integral; but you can still show that the integral is convergent using the comparison theorem. Use the integrand from part (b) as a comparison function to show that \(\int_{0}^{\infty} p(t) d t\) still converges.

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int_{1}^{4} \ln (\sqrt{x}+1) d x $$

We will discuss alternatives to comparing powers of \(x\) for finding the coefficients in a partial fraction expansion when the denominator polynomial has a repeated root. (a) Consider the rational function $$f(x)=\frac{x+3}{(x-1)(x+1)^{2}}$$ whose partial fraction expression is of the form $$f(x)=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{(x+1)^{2}}$$ for some set of constants \(A, B, C\) that need to be determined. To calculate these constants put all of the terms over a common denominator. \(\frac{x+3}{(x+1)^{2}(x-1)}=\frac{A(x+1)^{2}+B(x-1)(x+1)+C(x-1)}{(x-1)(x+1)^{2}}\) Explain why $$x+3=A(x+1)^{2}+B(x-1)(x+1)+C(x-1)$$ (b) One method to calculate \(A, B, C\) from \((7.24)\) is to substitute in specific values of \(x\). We showed in this section that some good choices are values that make one or more terms vanish. Show by substituting in \(x=1\) and \(x=-1\) that \(A=1\) and \(C=-1\). (c) Explain why there is no value of \(x\) that will make both the \(A(x+1)^{2}\) and \(C(x-1)\) terms vanish, without causing the \(B(x-1)(x+1)\) term to vanish. (d) Although we cannot isolate the term in \(B\), we can obtain more equations by substituting different values of \(x .\) By letting \(x=0\), show that: $$3=A-B-C$$ and use this equation to calculate \(B\). (e) Use your answers from (a)-(d) to evaluate $$\int \frac{x+3}{(x+1)^{2}(x-1)} d x$$ (f) Use the method given in parts (b) and (d) to find the partial fraction expansion of \(f(x)=\frac{3 x^{2}-2 x-4}{2 x^{2}(1+x)}.\)

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{1}^{5} x e^{-x^{2}} d x $$

The time between forest fires is often modeled using a Weibull distribution. According to this distribution the likelihood that a time \(t\) elapses between the end of one forest fire and the start of the next one is proportional to \(p(t)=\) \(t^{k-1} \exp \left(-t^{k}\right)\) where \(k\) is a positive constant. It can be shown using the laws of probability (see Chapter 12) that the probability of a second forest fire starting at any time following the first is proportional to \(\int_{0}^{\infty} p(t) d t\). (a) Assuming \(k=1\), show that \(\int_{0}^{\infty} p(t) d t\) is convergent and calculate the value of the integral. (b) Now assume \(k=2\). Show that \(\int_{0}^{\infty} p(t) d t\) is convergent and calculate the value of the integral (Hint: you will need to use integration by substitution). (c) Now adapt your argument from (b) to show that the integral is convergent for any value of \(k\) obeying \(k \geq 1\).

In Problems \(39-48\), first make an appropriate substitution and then use integration by parts to evaluate each integral. $$ \int_{0}^{1} x^{3} \ln \left(x^{2}+1\right) d x $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{\ln 4}^{\ln 7} \frac{e^{x}}{\left(e^{x}+1\right)^{2}} d x $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int x e^{-2 x} d x $$

Find the partial fraction expansion for each of the following functions. $$ f(x)=\frac{x^{3}-x^{2}+x-4}{\left(x^{2}+1\right)\left(x^{2}+4\right)} $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{\pi / 3} \sin x \cos x d x $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int x e^{-2 x^{2}} d x $$

Find the partial fraction expansion for each of the following functions. $$ f(x)=\frac{x^{3} \quad 3 x^{2}+x \quad 6}{\left(x^{2}+2\right)\left(x^{2}+1\right)} $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{-\pi / 6}^{\pi / 6} \sin ^{2} x \cos x d x $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int x(x+1)^{1 / 3} d x $$

Find the partial fraction expansion for each of the following functions. $$ f(x)=\frac{x^{2}+2 x-1}{(x-1)\left(x^{2}+1\right)} $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{\pi / 4} \tan x \sec ^{2} x d x $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int \frac{x}{(1-x)^{1 / 4}} d x $$

Find the partial fraction expansion for each of the following functions. $$ f(x)=\frac{x^{2}-x+6}{(x-2)\left(x^{2}+4\right)} $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{\pi / 4} \frac{\sin x}{\cos ^{2} x} d x $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int 2 x \sin \left(x^{2}\right) d x $$

Find the partial fraction expansion for each of the following functions. $$ f(x)=\frac{x^{3}+2 x^{2}+x+1}{x^{2}\left(x^{2}+1\right)} $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{2} \frac{x}{x+2} d x $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int 2 x^{2} \sin x d x $$

Find the partial fraction expansion for each of the following functions. $$ f(x)=\frac{x^{3}-3 x}{(x-1)^{2}\left(x^{2}+1\right)} $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{4}^{9} \frac{x}{x-3} d x $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{16+x^{2}} d x $$

Complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}-2 x+2} d x $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{e}^{e^{2}} \frac{d x}{x(\ln x)^{2}} $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int \frac{x}{x^{2}+5} d x $$

Complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}+4 x+5} d x $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{1}^{2} \frac{x d x}{\left(x^{2}+1\right) \ln \left(x^{2}+1\right)} $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int\left(\frac{\tan ^{2} x+1}{\tan x+1}\right) d x $$

Complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}-4 x+13} d x $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{1} x^{2} \sqrt{x^{3}+1} d x $$

In Problems 49-60, use either substitution or integration by parts to evaluate each integral. $$ \int(\sin x+1)^{2} \cos x d x $$

Complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}+2 x+5} d x $$

In Problems 43-58, use substitution to evaluate each definite integral. $$ \int_{0}^{2} x \sqrt{4-x^{2}} d x $$