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Subsection 1.9.2 Exercises

Recall that we are using \(\log x\) to denote the logarithm of \(x\) with base \(e\text{.}\) In other courses it is often denoted \(\ln x\text{.}\)

Exercises — Stage 1

1 (✳)

For each of the following integrals, choose the substitution that is most beneficial for evaluating the integral.

  1. \(\displaystyle \int \frac{2x^2}{\sqrt{9x^2-16}} \, \dee{x}\)
  2. \(\displaystyle \int \frac{x^4-3}{\sqrt{1-4x^2}} \, \dee{x}\)
  3. \(\displaystyle \int {(25+x^2)}^{-5/2} \, \dee{x}\)
2

For each of the following integrals, choose a trigonometric substitution that will eliminate the roots.

  1. \(\displaystyle\int \dfrac{1}{\sqrt{x^2-4x+1}}\dee{x}\)
  2. \(\displaystyle\int \dfrac{(x-1)^6}{(-x^2+2x+4)^{3/2}}\dee{x}\)
  3. \(\displaystyle\int \dfrac{1}{\sqrt{4x^2+6x+10}}\dee{x}\)
  4. \(\displaystyle\int \sqrt{x^2-x}\dee{x}\)
3

In each part of this question, assume \(\theta\) is an angle in the interval \(\left[ 0,\pi/2\right]\text{.}\)

  1. If \(\sin\theta=\dfrac{1}{20}\text{,}\) what is \(\cos\theta\) ?
  2. If \(\tan\theta=7\text{,}\) what is \(\csc\theta\) ?
  3. If \(\sec\theta=\dfrac{\sqrt{x-1}}{2}\text{,}\) what is \(\tan\theta\) ?
4

Simplify the following expressions.

  1. \(\sin\left(\arccos \left(\frac{x}{2}\right)\right)\)
  2. \(\sin\left(\arctan \left(\frac{1}{\sqrt{3}}\right)\right)\)
  3. \(\sec\left(\arcsin \left(\sqrt{x}\right)\right)\)

Exercises — Stage 2

5 (✳)

Evaluate \(\displaystyle\int \frac1{(x^2+4)^{3/2}} \,\dee{x}.\)

6 (✳)

Evaluate \(\displaystyle\int_0^4 \frac{1}{{(4+x^2)}^{3/2}}\,\dee{x}\text{.}\) Your answer may not contain inverse trigonometric functions.

7 (✳)

Evaluate \(\displaystyle\int_0^{5/2} \frac{\dee{x}}{\sqrt{25-x^2}}\text{.}\)

8 (✳)

Evaluate \(\displaystyle\int \frac{\dee{x}}{\sqrt{x^2+25}}\text{.}\) You may use that \({\displaystyle\int} \sec x\ \dee{x} = \log\big|\sec x+\tan x\big|+C\text{.}\)

9

Evaluate \(\displaystyle\int\frac{x+1}{\sqrt{2x^2+4x}} \, \dee{x}\text{.}\)

10 (✳)

Evaluate \(\displaystyle\int\frac{\dee{x}}{x^2\sqrt{x^2+16}}\text{.}\)

11 (✳)

Evaluate \(\displaystyle\int \frac{\dee{x}}{x^2\sqrt{x^2-9}}\) for \(x \ge 3\text{.}\) Do not include any inverse trigonometric functions in your answer.

12 (✳)

(a) Show that \(\displaystyle\int_0^{\pi/4}\cos^4\theta\dee{\theta}=(8+3\pi)/32\text{.}\)

(b) Evaluate \(\displaystyle\int_{-1}^1\frac{\dee{x}}{{(x^2+1)}^3}\text{.}\)

13

Evaluate \(\displaystyle\int_{-\pi/12}^{\pi/12} \dfrac{15x^3}{(x^2+1)(9-x^2)^{5/2}}\dee{x}\text{.}\)

14 (✳)

Evaluate \({\displaystyle\int} \sqrt{4-x^2}\,\dee{x}\text{.}\)

15 (✳)

Evaluate \(\displaystyle\int \frac{\sqrt{25x^2-4}}{x}\,\dee{x}\) for \(x\gt \frac{2}{5}\text{.}\)

16

Evaluate \(\displaystyle\int_{\sqrt{10}}^{\sqrt{17}} \frac{x^3}{\sqrt{x^2-1}}\, \dee{x}\text{.}\)

17 (✳)

Evaluate \(\displaystyle\int \frac{\dee{x}}{\sqrt{3-2x-x^2}}\text{.}\)

18

Evaluate \(\displaystyle\int \dfrac{1}{(2x-3)^3\sqrt{4x^2-12x+8}}\dee{x}\) for \(x \gt 2\text{.}\)

19

Evaluate \(\displaystyle\int_0^1\dfrac{x^2}{(x^2+1)^{3/2}}\dee{x}\text{.}\)

You may use that \(\int \sec x\dee{x} = \log|\sec x+\tan x| +C\text{.}\)

20

Evaluate \(\displaystyle\int \frac{1}{(x^2+1)^2}\dee{x}\text{.}\)

Exercises — Stage 3

21

Evaluate \(\displaystyle\int \dfrac{x^2}{\sqrt{x^2-2x+2}}\dee{x}\text{.}\)

You may assume without proof that \(\displaystyle\int \sec^3 \theta\dee{\theta} = \frac{1}{2}\sec\theta\tan\theta + \frac{1}{2}\log|\sec\theta+\tan\theta|+C\text{.}\)

22

Evaluate \(\displaystyle\int \dfrac{1}{\sqrt{3x^2+5x}}\dee{x}\text{.}\)

You may use that \(\int \sec x\dee{x} = \log|\sec x+\tan x| +C\text{.}\)

23

Evaluate \(\displaystyle\int\dfrac{(1+x^2)^{3/2}}{x}\dee{x}\text{.}\) You may use the fact that \(\displaystyle\int \csc \theta\dee{\theta}=\log|\cot \theta - \csc \theta|+C\text{.}\)

24

Below is the graph of the ellipse \(\left(\frac{x}{4}\right)^2+\left(\frac{y}{2}\right)^2=1\text{.}\) Find the area of the shaded region using the ideas from this section.

25

Let \(f(x) = \dfrac{|x|}{\sqrt[4]{1-x^2}}\text{,}\) and let \(R\) be the region between \(f(x)\) and the \(x\)-axis over the interval \([-\frac{1}{2},\frac{1}{2}]\text{.}\)

  1. Find the area of \(R\text{.}\)
  2. Find the volume of the solid formed by rotating \(R\) about the \(x\)-axis.
26

Evaluate \(\displaystyle\int \sqrt{1+e^x}\dee{x}\text{.}\) You may use the antiderivative \(\displaystyle\int \csc \theta \dee{\theta} = \log|\cot \theta - \csc \theta|+C\text{.}\)

27

Consider the following work.

\begin{align*} \int \frac{1}{1-x^2}\dee{x}&=\int\dfrac{1}{1-\sin^2 \theta}\cos\theta\dee{\theta} \qquad \mbox{using } x=\sin\theta, \dee{x}=\cos\theta\dee{\theta}\\ &=\int \frac{\cos \theta}{\cos^2 \theta}\dee\theta\\ &=\int \sec \theta\dee{\theta}\\ &=\log|\sec \theta + \tan \theta| +C \qquad\qquad\qquad \mbox{Example }\knowl{./knowl/eg_TRGINTopta.html}{\text{1.8.19}}\\ &=\log\left | \dfrac{1}{\sqrt{1-x^2}}+\dfrac{x}{\sqrt{1-x^2}} \right| +C \qquad \text{using the triangle below}\\ &=\log\left | \dfrac{1+x}{\sqrt{1-x^2}} \right| +C \end{align*}
  1. Differentiate \(\log\left| \dfrac{1+x}{\sqrt{1-x^2}}\right|\text{.}\)
  2. True or false: \(\displaystyle\int_{2}^{3} \frac{1}{1-x^2}\dee{x} = \left[\log\left| \dfrac{1+x}{\sqrt{1-x^2}}\right|\right]_{x=2}^{x=3}\)
  3. Was the work in the question correct? Explain.
28
  1. Suppose we are evaluating an integral that contains the term \(\sqrt{a^2-x^2}\text{,}\) where \(a\) is a positive constant, and we use the substitution \(x=a\sin u\) (with inverse \(u = \arcsin(x/a)\)), so that
    \begin{equation*} \sqrt{a^2-x^2} = \sqrt{a^2\cos^2u}= |a\cos u| \end{equation*}
    Under what circumstances is \(|a\cos u|\neq a\cos u\text{?}\)
  2. Suppose we are evaluating an integral that contains the term \(\sqrt{a^2+x^2}\text{,}\) where \(a\) is a positive constant, and we use the substitution \(x=a\tan u\) (with inverse \(u = \arctan(x/a)\)), so that
    \begin{equation*} \sqrt{a^2+x^2} = \sqrt{a^2\sec^2u}= |a\sec u| \end{equation*}
    Under what circumstances is \(|a\sec u|\neq a\sec u\text{?}\)
  3. Suppose we are evaluating an integral that contains the term \(\sqrt{x^2-a^2}\text{,}\) where \(a\) is a positive constant, and we use the substitution \(x=a\sec u\) (with inverse \(u = \arcsec(x/a)=\arccos(a/x)\)), so that
    \begin{equation*} \sqrt{x^2-a^2} = \sqrt{a^2\tan^2u}= |a\tan u| \end{equation*}
    Under what circumstances is \(|a\tan u|\neq a\tan u\text{?}\)
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