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Thomas Calculus

George B. Thomas, Jr.

Chapter 8

Techniques of Integration

Educators


Problem 1

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{0}^{1} \frac{16 x}{8 x^{2}+2} d x
$$

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Problem 2

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{x^{2}}{x^{2}+1} d x
$$

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Problem 3

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int(\sec x-\tan x)^{2} d x
$$

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Problem 4

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{\pi / 4}^{\pi / 3} \frac{d x}{\cos ^{2} x \tan x}
$$

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Problem 5

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{1-x}{\sqrt{1-x^{2}}} d x
$$

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Problem 6

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d x}{x-\sqrt{x}}
$$

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Problem 7

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{e^{-\cot z}}{\sin ^{2} z} d z
$$

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Problem 8

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{2^{\ln z^{3}}}{16 z} d z
$$

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Problem 9

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d z}{e^{z}+e^{-z}}
$$

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Problem 10

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{1}^{2} \frac{8 d x}{x^{2}-2 x+2}
$$

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Problem 11

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{-1}^{0} \frac{4 d x}{1+(2 x+1)^{2}}
$$

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Problem 12

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{-1}^{3} \frac{4 x^{2}-7}{2 x+3} d x
$$

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Problem 13

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d t}{1-\sec t}
$$

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Problem 14

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \csc t \sin 3 t d t
$$

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Problem 15

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{0}^{\pi / 4} \frac{1+\sin \theta}{\cos ^{2} \theta} d \theta
$$

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Problem 16

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d \theta}{\sqrt{2 \theta-\theta^{2}}}
$$

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Problem 17

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{\ln y}{y+4 y \ln ^{2} y} d y
$$

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Problem 18

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{2^{\sqrt{y}} d y}{2 \sqrt{y}}
$$

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Problem 19

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d \theta}{\sec \theta+\tan \theta}
$$

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Problem 20

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d t}{t \sqrt{3+t^{2}}}
$$

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Problem 21

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{4 t^{3}-t^{2}+16 t}{t^{2}+4} d t
$$

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Problem 22

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{x+2 \sqrt{x-1}}{2 x \sqrt{x-1}} d x
$$

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Problem 23

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{0}^{\pi / 2} \sqrt{1-\cos \theta} d \theta
$$

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Problem 24

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int(\sec t+\cot t)^{2} d t
$$

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Problem 25

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d y}{\sqrt{e^{2 y}-1}}
$$

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Problem 26

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{6 d y}{\sqrt{y}(1+y)}
$$

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Problem 27

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{2 d x}{x \sqrt{1-4 \ln ^{2} x}}
$$

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Problem 28

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d x}{(x-2) \sqrt{x^{2}-4 x+3}}
$$

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Problem 29

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int(\csc x-\sec x)(\sin x+\cos x) d x
$$

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Problem 30

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int 3 \sinh \left(\frac{x}{2}+\ln 5\right) d x
$$

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Problem 31

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{\sqrt{2}}^{3} \frac{2 x^{3}}{x^{2}-1} d x
$$

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Problem 32

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{-1}^{1} \sqrt{1+x^{2}} \sin x d x
$$

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Problem 33

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int_{-1}^{0} \sqrt{\frac{1+y}{1-y}} d y
$$

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Problem 34

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int e^{z+e^{t}} d z
$$

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Problem 35

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{7 d x}{(x-1) \sqrt{x^{2}-2 x-48}}
$$

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Problem 36

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d x}{(2 x+1) \sqrt{4 x+4 x^{2}}}
$$

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Problem 37

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{2 \theta^{3}-7 \theta^{2}+7 \theta}{2 \theta-5} d \theta
$$

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Problem 38

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\int \frac{d \theta}{\cos \theta-1}
$$

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Problem 39

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\begin{array}{l}{\int \frac{d x}{1+e^{x}}} \\ {\text {Hint: Use long division. }}\end{array}
$$

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Problem 40

The integrals in Exercises $1-40$ are in no particular order. Evaluate
each integral using any algebraic method or trigonometric identity
you think is appropriate, and then use a substitution to reduce it to a
standard form.
$$
\begin{array}{l}{\int \frac{\sqrt{x}}{1+x^{3}} d x} \\ {\text {Hint} : \text { Let } u=x^{3 / 2}}\end{array}
$$

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Problem 41

Area Find the area of the region bounded above by $y=2 \cos x$
and below by $y=\sec x,-\pi / 4 \leq x \leq \pi / 4$

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Problem 42

Volume Find the volume of the solid generated by revolving
the region in Exercise 41 about the $x$ -axis.

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Problem 43

Arc length Find the length of the curve $y=\ln (\cos x)$
$0 \leq x \leq \pi / 3$

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Problem 44

Are length Find the length of the curve $y=\ln (\sec x)$
$0 \leq x \leq \pi / 4$

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Problem 45

Centroid Find the centroid of the region bounded by the $x$ -axis,
the curve $y=\sec x,$ and the lines $x=-\pi / 4, x=\pi / 4$

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Problem 46

Centroid Find the centroid of the region bounded by the $x$ -axis,
the curve $y=\csc x,$ and the lines $x=\pi / 6, x=5 \pi / 6 .$

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Problem 47

The functions $y=e^{x^{3}}$ and $y=x^{3} e^{x^{3}}$ do not have elementary anti-
derivatives, but $y=\left(1+3 x^{3}\right) e^{x^{3}}$ does.
Evaluate
$$
\int\left(1+3 x^{3}\right) e^{x^{3}} d x
$$

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Problem 48

Use the substitution $u=\tan x$ to evaluate the integral
$$
\int \frac{d x}{1+\sin ^{2} x}
$$

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Problem 49

Use the substitution $u=x^{4}+1$ to evaluate the integral
$$
\int x^{7} \sqrt{x^{4}+1} d x
$$

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Problem 50

Using different substitutions Show that the integral
$$
\int\left(\left(x^{2}-1\right)(x+1)\right)^{-2 / 3} d x
$$
can be evaluated with any of the following substitutions.
$$
\begin{array}{l}{\text { a. } u=1 /(x+1)} \\ {\text { b. } u=((x-1) /(x+1))^{k} \text { for } k=1,1 / 2,1 / 3,-1 / 3,-2 / 3} \\ {\quad \text { and }-1}\end{array}
$$$$
\begin{array}{ll}{\text { c. } u=\tan ^{-1} x} & {\text { d. } u=\tan ^{-1} \sqrt{x}} \\ {\text { e. } u=\tan ^{-1}((x-1) / 2)} & {\text { f. } u=\cos ^{-1} x} \\ {\text { g. } u=\cosh ^{-1} x}\end{array}
$$
What is the value of the integral?

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