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1.13: More Integration Examples

  • Page ID
    91717
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    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
    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
    2

    Evaluate \(\displaystyle\int_{0}^{\frac{\pi }{2}}\sin^{4}x\cos^{5}x \, d{x}\text{.}\)

    3

    Evaluate \(\displaystyle\int \sqrt{3-5x^2}\, d{x}\text{.}\)

    4

    Evaluate \(\displaystyle\int_0^\infty \dfrac{x-1}{e^x}\, d{x}\text{.}\)

    5

    Evaluate \(\displaystyle\int \frac{-2}{3x^2+4x+1}\, d{x}\text{.}\)

    6

    Evaluate \(\displaystyle\int_1^2 x^2\log x \, d{x}\text{.}\)

    7 (✳)

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

    8 (✳)

    Evaluate the following integrals.

    1. \(\displaystyle\int_0^4\frac{x}{\sqrt{9+x^2}}\,\, d{x}\)
    2. \(\displaystyle\int_0^{\pi/2}\cos^3x\ \sin^2x\,\, d{x}\)
    3. \(\displaystyle\int_1^{e}x^3\log x\,\, d{x}\)
    9 (✳)

    Evaluate the following integrals.

    1. \(\displaystyle\int_0^{\pi/2} x\sin x\,\, d{x} \)
    2. \(\displaystyle\int_0^{\pi/2} \cos^5 x\,\, d{x} \)
    10 (✳)

    Evaluate the following integrals.

    1. \(\displaystyle\int_0^2 xe^x\,\, d{x}\)
    2. \(\displaystyle\int_0^1\frac{1}{\sqrt{1+x^2}}\,\, d{x}\)
    3. \(\displaystyle\int_3^5\frac{4x}{(x^2-1)(x^2+1)}\,\, d{x}\)
    11 (✳)

    Calculate the following integrals.

    1. \(\displaystyle\int_0^3\sqrt{9-x^2}\,\, d{x}\)
    2. \(\displaystyle\int_0^1\log(1+x^2)\,\, d{x}\)
    3. \(\displaystyle\int_3^\infty\frac{x}{(x-1)^2(x-2)}\,\, d{x}\)
    12

    Evaluate \(\displaystyle\int\frac{\sin^4\theta-5\sin^3\theta+4\sin^2\theta+10\sin\theta}{\sin^2\theta-5\sin\theta+6}\cos\theta\, d{\theta}\text{.}\)

    13 (✳)

    Evaluate the following integrals. Show your work.

    1. \(\displaystyle\int_0^{\pi\over 4}\sin^2(2x)\cos^3(2x)\ \, d{x}\)
    2. \(\displaystyle\int\big(9+x^2\big)^{-{3\over 2}}\ \, d{x}\)
    3. \(\displaystyle\int\frac{\, d{x}}{(x-1)(x^2+1)}\)
    4. \(\displaystyle\int x\arctan x\ \, d{x}\)
    14 (✳)

    Evaluate the following integrals.

    1. \(\displaystyle\int_0^{\pi/4}\sin^5(2x)\,\cos(2x)\ \, d{x}\)
    2. \(\displaystyle\int\sqrt{4-x^2}\ \, d{x}\)
    3. \(\displaystyle\int\frac{x+1}{x^2(x-1)}\ \, d{x}\)
    15 (✳)

    Calculate the following integrals.

    1. \(\displaystyle\int_0^\infty e^{-x} \sin(2x)\,\, d{x}\)
    2. \(\displaystyle\int_0^{\sqrt{2}}\frac{1}{(2+x^2)^{3/2}}\,\, d{x}\)
    3. \(\displaystyle\int_0^1 x\log(1+x^2)\,\, d{x}\)
    4. \(\displaystyle\int_3^\infty\frac{1}{(x-1)^2(x-2)}\,\, d{x}\)
    16 (✳)

    Evaluate the following integrals.

    1. \(\displaystyle\int x\,\log x\ \, d{x}\)
    2. \(\displaystyle\int\frac{(x-1)\,\, d{x}}{x^2+4x+5}\)
    3. \(\displaystyle\int\frac{\, d{x}}{x^2-4x+3}\)
    4. \(\displaystyle\int\frac{x^2\,\, d{x}}{1+x^6}\)
    17 (✳)

    Evaluate the following integrals.

    1. \(\displaystyle\int_0^1\arctan x\ \, d{x}\text{.}\)
    2. \(\displaystyle\int\frac{2x-1}{x^2-2x+5}\ \, d{x}\text{.}\)(✳)
    18 (✳)
    1. Evaluate \({\displaystyle \int\frac{x^2}{(x^3 + 1)^{101}}\,\, d{x}}\text{.}\)
    2. Evaluate \(\displaystyle\int \cos^3\!x\ \sin^4\!x\ \, d{x}\text{.}\)
    19 (✳)

    Evaluate \(\displaystyle\int_{\pi/2}^\pi \frac{\cos x}{\sqrt{\sin x}}\, d{x}\text{.}\)

    20 (✳)

    Evaluate the following integrals.

    1. \(\displaystyle\int \frac{e^x}{(e^x+1)(e^x-3)}\, \, d{x}\)
    2. \(\displaystyle\int_2^4 \frac{x^2-4x+4}{\sqrt{12+4x-x^2}}\, \, d{x}\)
    21 (✳)

    Evaluate these integrals.

    1. \(\displaystyle\int\frac{\sin^3x}{\cos^3x} \ \, d{x}\)
    2. \(\displaystyle\int_{-2}^{2}\frac{x^4}{x^{10}+16}\ \, d{x}\)
    22

    Evaluate \(\displaystyle\int x\sqrt{x-1}\, d{x}\text{.}\)

    23

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

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

    24

    Evaluate \(\displaystyle\int_0^{\pi/4} \sec^4x\tan^5x\,\, d{x}\text{.}\)

    25

    Evaluate \(\displaystyle\int \frac{3x^2+4x+6}{(x+1)^3} \, \, d{x}\text{.}\)

    26

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

    27

    Evaluate \(\displaystyle\int \sin x \cos x \tan x\, d{x}\text{.}\)

    28

    Evaluate \(\displaystyle\int \frac{1}{x^3+1}\, d{x}\text{.}\)

    29

    Evaluate \(\displaystyle\int (3x)^2\arcsin x \, d{x}\text{.}\)

    Stage 3
    30

    Evaluate \(\displaystyle\int_0^{\pi/2}\sqrt{\cos t+1}\ \, d{t}\text{.}\)

    31

    Evaluate \(\displaystyle\int_{0}^{e} \frac{\log\sqrt{x}}{x}\, d{x}\text{.}\)

    32

    Evaluate \(\displaystyle\int_{0.1}^{0.2} \frac{\tan x}{\log(\cos x)}\, \, d{x}\text{.}\)

    33 (✳)

    Evaluate these integrals.

    1. \(\displaystyle\int\sin(\log x) \ \, d{x}\)
    2. \(\displaystyle\int_0^1\frac{1}{x^2-5x+6}\ \, d{x}\)
    34 (✳)

    Evaluate (with justification).

    1. \(\displaystyle\int_0^3(x+1)\sqrt{9-x^2} \ \, d{x}\)
    2. \(\displaystyle\int\frac{4x+8}{(x-2)(x^2+4)}\ \, d{x}\)
    3. \(\displaystyle\int_{-\infty}^{+\infty} \frac{1}{e^x+e^{-x}}\ \, d{x}\)
    35

    Evaluate \(\displaystyle\int \sqrt{\frac{x}{1-x}}\, d{x}\text{.}\)

    36

    Evaluate \(\displaystyle\int_0^1e^{2x}e^{e^x}\,\, d{x}\text{.}\)

    37

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

    38

    Evaluate \(\displaystyle\int \frac{x\sin x}{\cos^2 x}\,\, d{x}\text{.}\)

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

    39

    Evaluate \(\displaystyle\int x(x+a)^n\, d{x}\text{,}\) where \(a\) and \(n\) are constants.

    40

    Evaluate \(\displaystyle\int\arctan (x^2)\, d{x}\text{.}\)


    This page titled 1.13: More Integration Examples is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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