More Integration Examples

Section 1.13 More Integration Examples

Exercises Exercises

Recall that we are using

\log x

to denote the logarithm of

x

with base

e .

In other courses it is often denoted

\ln x .

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Exercises — Stage 1 .

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1.

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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

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Exercises — Stage 2 .

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2.

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Evaluate

\int_{0}^{π / 2} \sin^{4} x \cos^{5} x d x .

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3.

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Evaluate

\int \sqrt{3 - 5 x^{2}} d x .

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4.

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Evaluate

\int_{0}^{\infty} \frac{x - 1}{e^{x}} d x .

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5.

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Evaluate

\int \frac{- 2}{3 x^{2} + 4 x + 1} d x .

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6.

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Evaluate

\int_{1}^{2} x^{2} \log x d x .

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7. (✳).

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Evaluate

\int \frac{x}{x^{2} - 3} d x .

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8. (✳).

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Evaluate the following integrals.

$\int_{0}^{4} \frac{x}{\sqrt{9 + x^{2}}} d x$
$\int_{0}^{π / 2} \cos^{3} x \sin^{2} x d x$
$\int_{1}^{e} x^{3} \log x d x$

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9. (✳).

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Evaluate the following integrals.

$\int_{0}^{π / 2} x \sin x d x$
$\int_{0}^{π / 2} \cos^{5} x d x$

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10. (✳).

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Evaluate the following integrals.

$\int_{0}^{2} x e^{x} d x$
$\int_{0}^{1} \frac{1}{\sqrt{1 + x^{2}}} d x$
$\int_{3}^{5} \frac{4 x}{(x^{2} - 1) (x^{2} + 1)} d x$

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11. (✳).

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Calculate the following integrals.

$\int_{0}^{3} \sqrt{9 - x^{2}} d x$
$\int_{0}^{1} \log (1 + x^{2}) d x$
$\int_{3}^{\infty} \frac{x}{(x - 1)^{2} (x - 2)} d x$

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12.

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Evaluate

\int \frac{\sin^{4} θ - 5 \sin^{3} θ + 4 \sin^{2} θ + 10 \sin θ}{\sin^{2} θ - 5 \sin θ + 6} \cos θ d θ .

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13. (✳).

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Evaluate the following integrals. Show your work.

$\int_{0}^{\frac{π}{4}} \sin^{2} (2 x) \cos^{3} (2 x) d x$
$\int (9 + x^{2})^{- \frac{3}{2}} d x$
$\int \frac{d x}{(x - 1) (x^{2} + 1)}$
$\int x \arctan x d x$

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14. (✳).

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Evaluate the following integrals.

$\int_{0}^{π / 4} \sin^{5} (2 x) \cos (2 x) d x$
$\int \sqrt{4 - x^{2}} d x$
$\int \frac{x + 1}{x^{2} (x - 1)} d x$

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15. (✳).

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Calculate the following integrals.

$\int_{0}^{\infty} e^{- x} \sin (2 x) d x$
$\int_{0}^{\sqrt{2}} \frac{1}{(2 + x^{2})^{3 / 2}} d x$
$\int_{0}^{1} x \log (1 + x^{2}) d x$
$\int_{3}^{\infty} \frac{1}{(x - 1)^{2} (x - 2)} d x$

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16. (✳).

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Evaluate the following integrals.

$\int x \log x d x$
$\int \frac{(x - 1) d x}{x^{2} + 4 x + 5}$
$\int \frac{d x}{x^{2} - 4 x + 3}$
$\int \frac{x^{2} d x}{1 + x^{6}}$

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17. (✳).

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Evaluate the following integrals.

$\int_{0}^{1} \arctan x d x .$
$\int \frac{2 x - 1}{x^{2} - 2 x + 5} d x .$

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18. (✳).

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Evaluate $\int \frac{x^{2}}{(x^{3} + 1)^{101}} d x .$
Evaluate $\int \cos^{3} x \sin^{4} x d x .$

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19.

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Evaluate

\int_{π / 2}^{π} \frac{\cos x}{\sqrt{\sin x}} d x .

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20. (✳).

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Evaluate the following integrals.

$\int \frac{e^{x}}{(e^{x} + 1) (e^{x} - 3)} d x$
$\int_{2}^{4} \frac{x^{2} - 4 x + 4}{\sqrt{12 + 4 x - x^{2}}} d x$

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21. (✳).

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Evaluate these integrals.

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$\int \frac{\sin^{3} x}{\cos^{3} x} d x$
$\int_{- 2}^{2} \frac{x^{4}}{x^{10} + 16} d x$

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22.

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Evaluate

\int x \sqrt{x - 1} d x .

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23.

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Evaluate

\int \frac{\sqrt{x^{2} - 2}}{x^{2}} d x

for

x \geq \sqrt{2} .

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You may use that

\int \sec x d x = \log | \sec x + \tan x | + C .

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24.

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Evaluate

\int_{0}^{π / 4} \sec^{4} x \tan^{5} x d x .

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25.

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Evaluate

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

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26.

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Evaluate

\int \frac{1}{x^{2} + x + 1} d x .

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27.

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Evaluate

\int \sin x \cos x \tan x d x .

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28.

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Evaluate

\int \frac{1}{x^{3} + 1} d x .

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29.

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Evaluate

\int (3 x)^{2} \arcsin x d x .

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Exercises — Stage 3 .

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30.

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Evaluate

\int_{0}^{π / 2} \sqrt{\cos t + 1} d t .

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31.

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Evaluate

\int_{1}^{e} \frac{\log \sqrt{x}}{x} d x .

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32.

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Evaluate

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

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33. (✳).

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Evaluate these integrals.

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$\int \sin (\log x) d x$
$\int_{0}^{1} \frac{1}{x^{2} - 5 x + 6} d x$

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34. (✳).

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Evaluate (with justification).

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$\int_{0}^{3} (x + 1) \sqrt{9 - x^{2}} d x$
$\int \frac{4 x + 8}{(x - 2) (x^{2} + 4)} d x$
$\int_{- \infty}^{+ \infty} \frac{1}{e^{x} + e^{- x}} d x$

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35.

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Evaluate

\int \sqrt{\frac{x}{1 - x}} d x .

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36.

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Evaluate

\int_{0}^{1} e^{2 x} e^{e^{x}} d x .

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37.

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Evaluate

\int \frac{x e^{x}}{(x + 1)^{2}} d x .

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38.

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Evaluate

\int \frac{x \sin x}{\cos^{2} x} d x .

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You may use that

\int \sec x d x = \log | \sec x + \tan x | + C .

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39.

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Evaluate

\int x (x + a)^{n} d x,

where

a

and

n

are constants.

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40.

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Evaluate

\int \arctan (x^{2}) d x .

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