Chapter 7

apply integration by parts twice to evaluate each integral. $$ \int \cos (\ln x) d x $$

Without solving the logistic equation or referring to its solution, explain how you know that if \(y_{0}

Perform the indicated integrations. $$ \int \frac{1}{x^{2}+2 x+5} d x $$

Consider the logistic equation with initial condition \(y(0)=y_{0}\). Assuming \(y_{0}

Perform the indicated integrations. $$ \int \frac{1}{x^{2}-4 x+9} d x $$

Find \(c\) so that \(\int_{0}^{c} \frac{1}{3 \sqrt{2 \pi}} x^{3 / 2} e^{-x / 2} d x=0.90\).

Perform the indicated integrations. $$ \int \frac{d x}{9 x^{2}+18 x+10} $$

Find \(c\) so that \(\int_{-c}^{c} \frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2} d x=0.95 .\) Hint: Use symmetry.

Perform the indicated integrations. $$ \int \frac{d x}{\sqrt{16+6 x-x^{2}}} $$

use integration by parts to derive the given formula. $$ \begin{array}{l} \int \cos 5 x \sin 7 x d x= \\ -\frac{7}{24} \cos 5 x \cos 7 x-\frac{5}{24} \sin 5 x \sin 7 x+C \end{array} $$

The Law of Mass Action in chemistry results in the differential equation $$ \frac{d x}{d t}=k(a-x)(b-x), \quad k>0, \quad a>0, \quad b>0 $$ where \(x\) is the amount of a substance at time \(t\) resulting from the reaction of two others. Assume that \(x=0\) when \(t=0\). (a) Solve this differential equation in the case \(b>a\). (b) Show that \(x \rightarrow a\) as \(t \rightarrow \infty\) (if \(b>a\) ). (c) Suppose that \(a=2\) and \(b=4\), and that 1 gram of the substance is formed in 20 minutes. How much will be present in 1 hour? (d) Solve the differential equation if \(a=b\).

Perform the indicated integrations. $$ \int \frac{x+1}{9 x^{2}+18 x+10} d x $$

use integration by parts to derive the given formula. $$ \int e^{\alpha z} \sin \beta z d z=\frac{e^{\alpha z}(\alpha \sin \beta z-\beta \cos \beta z)}{\alpha^{2}+\beta^{2}}+C $$

The differential equation $$ \frac{d y}{d t}=k(y-m)(M-y), y(0)=y_{0} $$ with \(k>0\) and \(0 \leq m

Perform the indicated integrations. $$ \int \frac{x+1}{9 x^{2}+18 x+10} d x $$

use integration by parts to derive the given formula. $$ \int e^{\alpha z} \cos \beta z d z=\frac{e^{\alpha z}(\alpha \cos \beta z+\beta \sin \beta z)}{\alpha^{2}+\beta^{2}}+C $$

As a model for the production of trypsin from trypsinogen in digestion, biochemists have proposed the model $$ \frac{d y}{d t}=k(A-y)(B+y) $$ where \(k>0, A\) is the initial amount of trypsinogen, and \(B\) is the original amount of trypsin. Solve this differential equation.

Perform the indicated integrations. $$ \int \frac{d t}{t \sqrt{2 t^{2}-9}} $$

use integration by parts to derive the given formula. $$ \int x^{\alpha} \ln x d x=\frac{x^{\alpha+1}}{\alpha+1} \ln x-\frac{x^{\alpha+1}}{(\alpha+1)^{2}}+C, \alpha \neq-1 $$

54\. Evaluate $$ \int_{\pi / 6}^{\pi / 2} \frac{\cos x}{\sin x\left(\sin ^{2} x+1\right)^{2}} d x $$

Perform the indicated integrations. $$ \int \frac{\tan x}{\sqrt{\sec ^{2} x-4}} d x $$

use integration by parts to derive the given formula. $$ \begin{array}{rl} \int x^{\alpha}(\ln x)^{2} & d x=\frac{x^{\alpha+1}}{\alpha+1}(\ln x)^{2} \\ \- & 2 \frac{x^{\alpha+1}}{(\alpha+1)^{2}} \ln x+2 \frac{x^{\alpha+1}}{(\alpha+1)^{3}}+C, \alpha \neq-1 \end{array} $$

Find the length of the curve \(y=\ln (\cos x)\) between \(x=0\) and \(x=\pi / 4\).

In Problems 55-61, derive the given reduction formula using integration by parts. $$ \int x^{\alpha} e^{\beta x} d x=\frac{x^{\alpha} e^{\beta x}}{\beta}-\frac{\alpha}{\beta} \int x^{\alpha-1} e^{\beta x} d x $$

derive the given reduction formula using integration by parts. $$ \int x^{\alpha} \sin \beta x d x=-\frac{x^{\alpha} \cos \beta x}{\beta}+\frac{\alpha}{\beta} \int x^{\alpha-1} \cos \beta x d x $$

Evaluate \(\int_{0}^{2 \pi} \frac{x|\sin x|}{1+\cos ^{2} x} d x .\) Hint: Make the substitution \(u=x-\pi\) in the definite integral and then use symmetry properties.

derive the given reduction formula using integration by parts. $$ \int x^{\alpha} \cos \beta x d x=\frac{x^{\alpha} \sin \beta x}{\beta}-\frac{\alpha}{\beta} \int x^{\alpha-1} \sin \beta x d x $$

Let \(R\) be the region bounded by \(y=\sin x\) and \(y=\cos x\) between \(x=-\pi / 4\) and \(x=3 \pi / 4\). Find the volume of the solid obtained when \(R\) is revolved about \(x=-\pi / 4 .\) Hint: Use cylindrical shells to write a single integral, make the substitution \(u=x-\pi / 4\), and apply symmetry properties.

derive the given reduction formula using integration by parts. $$ \int(\ln x)^{\alpha} d x=x(\ln x)^{\alpha}-\alpha \int(\ln x)^{\alpha-1} d x $$

Over what subintervals of \([0,2]\) is the Fresnel function \(C(x)\) increasing? Concave up?

derive the given reduction formula using integration by parts. $$ \int \cos ^{\alpha} x d x=\frac{\cos ^{\alpha-1} x \sin x}{\alpha}+\frac{\alpha-1}{\alpha} \int \cos ^{\alpha-2} x d x $$

derive the given reduction formula using integration by parts. $$ \begin{array}{l} \int \cos ^{\alpha} \beta x d x= \\ \quad \frac{\cos ^{\alpha-1} \beta x \sin \beta x}{\alpha \beta}+\frac{\alpha-1}{\alpha} \int \cos ^{\alpha-2} \beta x d x \end{array} $$

. Find the area of the region bounded by the curve \(y=\ln x\), the \(x\) -axis, and the line \(x=e .\)

. Find the area of the region bounded by the curves \(y=3 e^{-x / 3}, y=0, x=0\), and \(x=9 .\) Make a sketch.

Find the area of the region bounded by the graphs of \(y=x \sin x\) and \(y=x \cos x\) from \(x=0\) to \(x=\pi / 4\).

. Evaluate the integral \(\int \cot x \csc ^{2} x d x\) by parts in two different ways: (a) By differentiating \(\cot x\) (b) By differentiating \(\csc x\) (c) Show that the two results are equivalent up to a constant.

If \(p(x)\) is a polynomial of degree \(n\) and \(G_{1}, G_{2}, \ldots, G_{n+1}\), are successive antiderivatives of a function \(g\), then, by repeated integration by parts, \(\int p(x) g(x) d x=p(x) G_{1}(x)-p^{\prime}(x) G_{2}(x)+p^{\prime \prime}(x) G_{3}(x)-\cdots\) \(+(-1)^{n} p^{(n)}(x) G_{n+1}(x)+C\) Use this result to find each of the following: (a) \(\int\left(x^{3}-2 x\right) e^{x} d x\) (b) \(\int\left(x^{2}-3 x+1\right) \sin x d x\)

Find the error in the following "proof" that \(0=1 .\) In \(\int(1 / t) d t\), set \(u=1 / t\) and \(d v=d t .\) Then \(d u=-t^{-2} d t\) and \(u v=1\). Integration by parts gives $$ \int(1 / t) d t=1-\int(-1 / t) d t $$

. Suppose that you want to evaluate the integral $$ \int e^{5 x}(4 \cos 7 x+6 \sin 7 x) d x $$ and you know from experience that the result will be of the form \(e^{5 x}\left(C_{1} \cos 7 x+C_{2} \sin 7 x\right)+C_{3} .\) Compute \(C_{1}\) and \(C_{2}\) by differ- entiating the result and setting it equal to the integrand.

Show that $$ f(t)=f(a)+\sum_{i=1}^{n} \frac{f^{(i)}(a)}{i !}(t-a)^{i}+\int_{a}^{t} \frac{(t-x)^{n}}{n !} f^{(n+1)}(x) d x $$ provided that \(f\) can be differentiated \(n+1\) times.

The Beta function, which is important in many branches of mathematics, is defined as $$ B(\alpha, \beta)=\int_{0}^{1} x^{\alpha-1}(1-x)^{\beta-1} d x $$ with the condition that \(\alpha \geq 1\) and \(\beta \geq 1\). (a) Show by a change of variables that $$ B(\alpha, \beta)=\int_{0}^{1} x^{\beta-1}(1-x)^{\alpha-1} d x=B(\beta, \alpha) $$ (b) Integrate by parts to show that \(B(\alpha, \beta)=\frac{\alpha-1}{\beta} B(\alpha-1, \beta+1)=\frac{\beta-1}{\alpha} B(\alpha+1, \beta-1)\) (c) Assume now that \(\alpha=n\) and \(\beta=m\), and that \(n\) and \(m\) are positive integers. By using the result in part (b) repeatedly, show that $$ B(n, m)=\frac{(n-1) !(m-1) !}{(n+m-1) !} $$

. Suppose that \(f(t)\) has the property that \(f^{\prime}(a)=f^{\prime}(b)=0\) and that \(f(t)\) has two continuous derivatives. Use integration by parts to prove that \(\int_{a}^{b} f^{\prime \prime}(t) f(t) d t \leq 0 .\) Hint \(:\) Use integration by parts by differentiating \(f(t)\) and integrating \(f^{\prime \prime}(t) .\) This result has many applications in the field of applied mathematics.

Derive the formula $$ \int_{0}^{x}\left(\int_{0}^{t} f(z) d z\right) d t=\int_{0}^{x} f(t)(x-t) d t $$ using integration by parts.

If \(P_{n}(x)\) is a polynomial of degree \(n\), show that $$ \int e^{x} P_{n}(x) d x=e^{x} \sum_{j=0}^{n}(-1)^{j} \frac{d^{j} P_{n}(x)}{d x^{j}} $$