Deck 34: Methods of Integration

Question 1

Integrate: $\int 3 x^{2} e^{x^{3}} d x$

Answer 1

$e^{x^{3}}+C$

Question 2

Integrate: $\int \frac{d x}{e^{2 x}}$

Answer 2

$-\frac{1}{2 e^{2 x}}+C$

Question 3

Integrate: $\int_{}^{}\left(e^{2 x}+x e^{x^{2}}\right) \cdot dx$

Answer 3

$\frac{e^{2 x}+e^{x^{2}}}{2}+C$

Question 4

Evaluate:
$\int_{1}^{e} \frac{(\ln x)^{2} d x}{x}$

Question 5

Find the volume generated by rotating the first quadrant area bounded by the set of curves around the $x$ -axis: $y=3 e^{x}$ and $x=1$ . Round your answer to three significant digits.

Question 6

Integrate: $\int 4^{5 x} d x$

Question 7

Integrate: $\int_{0}^{2} x e^{x^{2}-1} d x$ . Round your answer to three significant digits.

Question 8

Integrate: $\int\left(e^{2 x}-e^{-2 x}\right) d x$

Question 9

Integrate: $\int \sqrt{7^{5 x}} d x$

Question 10

Integrate: $\int \log 7 x \cdot d x$

Question 11

Integrate: $\int 4 \log 2 x d x$

Question 12

Integrate: $\int_{1}^{4} \frac{18 x d x}{e^{3 x^{2}}}$

Question 13

Integrate: $\int(10 x-5) \ln \left(x^{2}-x\right) d x$

Question 14

Integrate: $\int\left(e^{\frac{x}{2}}-\pi\right)^{2} d x$

Question 15

Integrate: $\int_{\frac{1}{4}}^{1} \ln 4 x d x$

Question 16

Evaluate: $\int_{0}^{\pi / 2} \sin^{4} x \cos x d x$

Question 17

Evaluate: $\int_{0}^{\pi / 2} \cos x e^{\sin x} d x$ . Round your answer to three significant digits.

Question 18

Integrate: $\int \cos 4 y d y$

Question 19

Integrate: $\int \tan (1-3 x) d x$

Question 20

Integrate: $\int \csc 2 x d x$

Question 21

Integrate: $\int \tan x \sec^{4} x d x$

Question 22

Integrate: $\int \frac{\sin \sqrt{x}}{\sqrt{x}} \cdot d x$

Question 23

Find the area under one arch of the function: $y=\sin 2 x$

Question 24

Integrate: $\int x \cot \left(x^{2}+1\right) dx$

Question 25

Find the area under the curve: $y=\sin^{2} x \cos x+2 x$ from $x=0$ to $x=\pi$ . Round your answer to three significant digits.

Question 26

Find the volume of the solid generated by rotating the area between $y=\sin x$ and the $x$ -axis from $\mathrm{x}=0$ to $\mathrm{x}=\pi$ about the $\mathrm{x}$ -axis. Round your answer to three significant digits.

Question 27

Integrate: $\int x^{2} \sin x^{3} d x$

Question 28

Integrate: $\int(4 x+2) \cos \left(x^{2}+x\right) d x$

Question 29

Integrate: $\int e^{x} \sin e^{x} d x$

Question 30

Find the area bounded by the curve $y=2 \tan \frac{x}{4}$ and the $x$ -axis from $x=0$ to $x=\pi$ .

Question 31

Integrate: $\int x \tan 2 x d x$

Question 32

Integrate: $\int 4 x \sec x^{2} d x$

Question 33

Find the average ordinate for the function in the given interval: $y=\sqrt{3 x+1}$ from $x=1$ to $x=5$ . Round your answer to three significant digits.

Question 34

Find the rms value for the function in the given interval: $y=5 e^{-3 t}$ from $t=0$ to $t=1.5$

Question 35

Find the average ordinate for the function in the given interval: $y=5-x^{2}$ from $x=0$ to $x=2$ . Round your answer to three significant digits.

Question 36

Find the average ordinate for the function in the given interval: $y=\frac{x}{\sqrt{x^{2}-15}}$ from 4 to 8

Question 37

Find the rms value for the function in the given interval: $y=2 \sin 3 x$ from $x=0$ to $x=\frac{\pi}{3}$ . Round your answer to three significant digits.

Question 38

Find the average ordinate for the function $y=\sqrt{4-\frac{x^{2}}{4}}$ in the interval $x=-4$ to $x=4$ .

Question 39

Find the rms value for the function in the given interval: $y=5 \cos 4 x$ from $x=-\frac{\pi}{4}$ to $x=\frac{\pi}{2}$ . Round your answer to three significant digits.

Question 40

Find the average ordinate for the function $y=2 x^{2}-x^{3}$ in the interval $x=0$ to $x=2$ . Round your answer to three significant digits.

Question 41

Integrate by parts: $\int x \cos 2 x d x$

Question 42

Integrate by parts: $\int \sin^{-1} x d x$

Question 43

Integrate by parts: $\int \frac{x^{5} d x}{\sqrt{1-2 x^{3}}}$

Question 44

Integrate by parts: $\int 4 x^{2} e^{2 x} d x$

Question 45

Integrate by parts: $\int \sin^{3} x d x$

Question 46

Integrate by parts: $\int \ln \left(x^{2}+4\right) d x$

Question 47

Integrate by parts: $\int \tan^{-1} x d x$

Question 48

Integrate by parts: $\int x^{3} \ln x d x$

Question 49

Integrate by parts:
$\int_{0}^{4} 8 x e^{2 x} d x$

Question 50

Integrate by parts: $\int_{0}^{\frac{\pi}{2}} 4 x \cos 2 x d x$

Question 51

Integrate by parts: $\int e^{x} \sin x d x$

Question 52

Integrate: $\int \frac{x^{2}+4 x+2}{x^{2}+2} d x$

Question 53

Integrate: $\int_{-2}^{1} \frac{-4}{x^{2}+x-6} d x$ . Round your answer to three significant digits.

Question 54

Integrate: $\int \frac{4 x^{2}+3 x+9}{x^{3}-9 x} d x$

Question 55

Integrate: $\int \frac{2 x+4}{x^{3}-2 x^{2}} d x$

Question 56

Integrate: $\int \frac{-4 x-8}{2 x^{3}-x^{2}+2 x-1} d x$

Question 57

Integrate: $\int \frac{3 x-2}{x^{2}+3 x+2} d x$

Question 58

Integrate: Round your answer to three significant digits.

Question 59

Find the area bounded by the curve $y=\frac{2 x-1}{x^{2}+2 x}$ and the x-axis from $x=0.5$ to $x=10$ . Round your answer to three significant digits.

Question 60

Integrate: $\int \frac{4 x^{3}-8 x^{2}+5}{2 x-1} d x$

Question 61

Integrate: $\int \frac{3 x^{2}-2 x+27}{x^{3}+9 x} d x$

Question 62

Integrate: $\int \frac{2 x^{2}+18 x+20}{x^{3}+4 x^{2}+4 x} d x$

Question 63

Integrate: $\int \frac{\sqrt{x}}{1+\sqrt[3]{x}} d x$

Question 64

Integrate: $\int_{0}^{3} \frac{x^{2}}{\sqrt{x+1}} d x$ . Round your answer to three significant digits.

Question 65

Integrate: $\int \sqrt{1+e^{x}} d x$

Question 66

Integrate: $\int_{0}^{\sqrt{5}} \frac{x}{\sqrt{x^{2}+4}} d x$

Question 67

Integrate:
$\int \frac{8 x d x}{(2 x+1)^{3 / 2}}$

Question 68

Integrate: $\int \frac{4 x d x}{\sqrt{1-2 x}}$

Question 69

Integrate: $\int \frac{d x}{\sqrt{x}-\sqrt[4]{x}}$

Question 70

Integrate: $\int \frac{x^{2} d x}{\sqrt{x^{2}+4}}$

Question 71

Integrate: $\int \frac{x^{2} d x}{\sqrt{1-4 x}}$

Question 72

Integrate: $\int x \sqrt[3]{x+1} d x$

Question 73

Integrate: $\int \frac{d x}{\sqrt{4-x^{2}}}$

Question 74

Integrate: $\int \frac{d x}{\sqrt{x^{2}-25}}$

Question 75

Integrate: $\int \sqrt{x^{2}+9} d x$

Question 76

Integrate: $\int_{0.1}^{0.5} \frac{d x}{x^{2} \sqrt{1-x^{2}}}$

Question 77

Integrate: $\int \frac{x^{2} d x}{\sqrt{x^{2}-16}}$

Question 78

Integrate: $\int x^{2} \sqrt{x^{2}+1} d x$

Question 79

Evaluate the improper integral: $\int_{2}^{\infty} \frac{d x}{e^{x}}$ . Round your answer to three significant digits.

Question 80

Evaluate the improper integral: $\int_{3}^{\infty} \frac{d x}{x^{2}-1}$ . Round your answer to three significant digits.