1.13: More Integration Examples
- Page ID
- 91717
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(\ln x\text{.}\)
Stage 1
Match the integration method to a common kind of integrand it's used to antidifferentiate.
| (A) \(u=f(x)\) substitution | (I) | a function multiplied by its derivative |
| (B) trigonometric substitution | (II) | a polynomial times an exponential |
| (C) integration by parts | (III) | a rational function |
| (D) partial fractions | (IV) | the square root of a quadratic function |
Stage 2
Evaluate \(\displaystyle\int_{0}^{\frac{\pi }{2}}\sin^{4}x\cos^{5}x \, d{x}\text{.}\)
Evaluate \(\displaystyle\int \sqrt{3-5x^2}\, d{x}\text{.}\)
Evaluate \(\displaystyle\int_0^\infty \dfrac{x-1}{e^x}\, d{x}\text{.}\)
Evaluate \(\displaystyle\int \frac{-2}{3x^2+4x+1}\, d{x}\text{.}\)
Evaluate \(\displaystyle\int_1^2 x^2\log x \, d{x}\text{.}\)
Evaluate \(\displaystyle\int\frac{x}{x^2-3}\,\, d{x}\text{.}\)
Evaluate the following integrals.
- \(\displaystyle\int_0^4\frac{x}{\sqrt{9+x^2}}\,\, d{x}\)
- \(\displaystyle\int_0^{\pi/2}\cos^3x\ \sin^2x\,\, d{x}\)
- \(\displaystyle\int_1^{e}x^3\log x\,\, d{x}\)
Evaluate the following integrals.
- \(\displaystyle\int_0^{\pi/2} x\sin x\,\, d{x} \)
- \(\displaystyle\int_0^{\pi/2} \cos^5 x\,\, d{x} \)
Evaluate the following integrals.
- \(\displaystyle\int_0^2 xe^x\,\, d{x}\)
- \(\displaystyle\int_0^1\frac{1}{\sqrt{1+x^2}}\,\, d{x}\)
- \(\displaystyle\int_3^5\frac{4x}{(x^2-1)(x^2+1)}\,\, d{x}\)
Calculate the following integrals.
- \(\displaystyle\int_0^3\sqrt{9-x^2}\,\, d{x}\)
- \(\displaystyle\int_0^1\log(1+x^2)\,\, d{x}\)
- \(\displaystyle\int_3^\infty\frac{x}{(x-1)^2(x-2)}\,\, d{x}\)
Evaluate \(\displaystyle\int\frac{\sin^4\theta-5\sin^3\theta+4\sin^2\theta+10\sin\theta}{\sin^2\theta-5\sin\theta+6}\cos\theta\, d{\theta}\text{.}\)
Evaluate the following integrals. Show your work.
- \(\displaystyle\int_0^{\pi\over 4}\sin^2(2x)\cos^3(2x)\ \, d{x}\)
- \(\displaystyle\int\big(9+x^2\big)^{-{3\over 2}}\ \, d{x}\)
- \(\displaystyle\int\frac{\, d{x}}{(x-1)(x^2+1)}\)
- \(\displaystyle\int x\arctan x\ \, d{x}\)
Evaluate the following integrals.
- \(\displaystyle\int_0^{\pi/4}\sin^5(2x)\,\cos(2x)\ \, d{x}\)
- \(\displaystyle\int\sqrt{4-x^2}\ \, d{x}\)
- \(\displaystyle\int\frac{x+1}{x^2(x-1)}\ \, d{x}\)
Calculate the following integrals.
- \(\displaystyle\int_0^\infty e^{-x} \sin(2x)\,\, d{x}\)
- \(\displaystyle\int_0^{\sqrt{2}}\frac{1}{(2+x^2)^{3/2}}\,\, d{x}\)
- \(\displaystyle\int_0^1 x\log(1+x^2)\,\, d{x}\)
- \(\displaystyle\int_3^\infty\frac{1}{(x-1)^2(x-2)}\,\, d{x}\)
Evaluate the following integrals.
- \(\displaystyle\int x\,\log x\ \, d{x}\)
- \(\displaystyle\int\frac{(x-1)\,\, d{x}}{x^2+4x+5}\)
- \(\displaystyle\int\frac{\, d{x}}{x^2-4x+3}\)
- \(\displaystyle\int\frac{x^2\,\, d{x}}{1+x^6}\)
Evaluate the following integrals.
- \(\displaystyle\int_0^1\arctan x\ \, d{x}\text{.}\)
- \(\displaystyle\int\frac{2x-1}{x^2-2x+5}\ \, d{x}\text{.}\)(✳)
- Evaluate \({\displaystyle \int\frac{x^2}{(x^3 + 1)^{101}}\,\, d{x}}\text{.}\)
- Evaluate \(\displaystyle\int \cos^3\!x\ \sin^4\!x\ \, d{x}\text{.}\)
Evaluate \(\displaystyle\int_{\pi/2}^\pi \frac{\cos x}{\sqrt{\sin x}}\, d{x}\text{.}\)
Evaluate the following integrals.
- \(\displaystyle\int \frac{e^x}{(e^x+1)(e^x-3)}\, \, d{x}\)
- \(\displaystyle\int_2^4 \frac{x^2-4x+4}{\sqrt{12+4x-x^2}}\, \, d{x}\)
Evaluate these integrals.
- \(\displaystyle\int\frac{\sin^3x}{\cos^3x} \ \, d{x}\)
- \(\displaystyle\int_{-2}^{2}\frac{x^4}{x^{10}+16}\ \, d{x}\)
Evaluate \(\displaystyle\int x\sqrt{x-1}\, d{x}\text{.}\)
Evaluate \(\displaystyle\int \frac{\sqrt{x^2-2}}{x^2}\, d{x}\text{.}\)
You may use that \(\int \sec x\, d{x} = \log|\sec x+\tan x| +C\text{.}\)
Evaluate \(\displaystyle\int_0^{\pi/4} \sec^4x\tan^5x\,\, d{x}\text{.}\)
Evaluate \(\displaystyle\int \frac{3x^2+4x+6}{(x+1)^3} \, \, d{x}\text{.}\)
Evaluate \(\displaystyle\int\frac{1}{x^2+x+1}\,\, d{x}\text{.}\)
Evaluate \(\displaystyle\int \sin x \cos x \tan x\, d{x}\text{.}\)
Evaluate \(\displaystyle\int \frac{1}{x^3+1}\, d{x}\text{.}\)
Evaluate \(\displaystyle\int (3x)^2\arcsin x \, d{x}\text{.}\)
Stage 3
Evaluate \(\displaystyle\int_0^{\pi/2}\sqrt{\cos t+1}\ \, d{t}\text{.}\)
Evaluate \(\displaystyle\int_{0}^{e} \frac{\log\sqrt{x}}{x}\, d{x}\text{.}\)
Evaluate \(\displaystyle\int_{0.1}^{0.2} \frac{\tan x}{\log(\cos x)}\, \, d{x}\text{.}\)
Evaluate these integrals.
- \(\displaystyle\int\sin(\log x) \ \, d{x}\)
- \(\displaystyle\int_0^1\frac{1}{x^2-5x+6}\ \, d{x}\)
Evaluate (with justification).
- \(\displaystyle\int_0^3(x+1)\sqrt{9-x^2} \ \, d{x}\)
- \(\displaystyle\int\frac{4x+8}{(x-2)(x^2+4)}\ \, d{x}\)
- \(\displaystyle\int_{-\infty}^{+\infty} \frac{1}{e^x+e^{-x}}\ \, d{x}\)
Evaluate \(\displaystyle\int \sqrt{\frac{x}{1-x}}\, d{x}\text{.}\)
Evaluate \(\displaystyle\int_0^1e^{2x}e^{e^x}\,\, d{x}\text{.}\)
Evaluate \(\displaystyle\int\frac{xe^x}{(x+1)^2}\, d{x}\text{.}\)
Evaluate \(\displaystyle\int \frac{x\sin x}{\cos^2 x}\,\, d{x}\text{.}\)
You may use that \(\int \sec x\, d{x} = \log|\sec x+\tan x| +C\text{.}\)
Evaluate \(\displaystyle\int x(x+a)^n\, d{x}\text{,}\) where \(a\) and \(n\) are constants.
Evaluate \(\displaystyle\int\arctan (x^2)\, d{x}\text{.}\)