Integration

Question 1

What is the integration by parts formula?

Accepted Answer

$\int u \, dv = uv - \int v \, du$

Question 2

What is the standard substitution used for $\int \sqrt{a^2 - x^2} \, dx$?

Accepted Answer

$x = a \sin A \quad 	ext{or} \quad x = a \cos A$

Question 3

What is the standard substitution used for $\int \sqrt{x^2 - a^2} \, dx$?

Accepted Answer

$x = a \sin A \quad 	ext{or} \quad x = a \cos A$

Question 4

What is the standard substitution used for $\int \sqrt{a^2 + x^2} \, dx$?

Accepted Answer

$x = a 	an A \quad 	ext{or} \quad x = a \cot A$

Question 5

What is the standard substitution used for $\int \sqrt{\frac{a \pm x}{a \mp x}} \, dx$?

Accepted Answer

$x = a \cos 2A$

Question 6

What is the standard substitution used for $\int {\frac{a \pm x}{a \mp x}} \, dx$?

Accepted Answer

$x = a 	an A$

Question 7

$\int (ax + b)^n \, dx$

Accepted Answer

$\frac{1}{a} \cdot \frac{1}{n+1} (ax + b)^{n+1}$ + C

Question 8

$\int a^x \, dx$

Accepted Answer

$\frac{a^x}{\ln a}$ + C

Question 9

$\int \alpha^{ax+b} \, dx$

Accepted Answer

$\frac{1}{a \ln \alpha} \alpha^{ax+b}$ + C

Question 10

$\int e^{ax+b} \, dx$

Accepted Answer

$\frac{1}{a} e^{ax+b}$ + C

Question 11

$\int \frac{1}{ax+b} \, dx$

Accepted Answer

$\frac{1}{a} \ln|ax+b|$ + C

Question 12

$\int \sin(ax+b) \, dx$

Accepted Answer

$-\frac{1}{a} \cos(ax+b)$ + C

Question 13

$\int \cos(ax+b) \, dx$

Accepted Answer

$\frac{1}{a} \sin(ax+b)$ + C

Question 14

$\int \sec^2(ax+b) \, dx$

Accepted Answer

$\frac{1}{a} 	an(ax+b)$ + C

Question 15

$\int \frac{1}{\sqrt{1 - (ax+b)^2}} \, dx$

Accepted Answer

$\frac{1}{a} \arcsin(ax+b)$ + C

Question 16

$\int \frac{-1}{\sqrt{1 - (ax+b)^2}} \, dx$

Accepted Answer

$\frac{1}{a} \arccos(ax+b)$ + C

Question 17

$\int \frac{1}{1 + (ax+b)^2} \, dx$

Accepted Answer

$\frac{1}{a} \arctan(ax+b)$ + C

Question 18

$\int 	an(ax+b) \, dx$

Accepted Answer

$-\frac{1}{a} \ln(\cos(ax+b))$ + C

Question 19

$\int \sinh(ax+b) \, dx$

Accepted Answer

$\frac{1}{a} \cosh(ax+b)$ + C

Question 20

$\int \cosh(ax+b) \, dx$

Accepted Answer

$\frac{1}{a} \sinh(ax+b)$ + C

Question 21

$\int 	ext{sech}^2(ax+b) \, dx$

Accepted Answer

$\frac{1}{a} 	anh(ax+b)$ + C

Question 22

$\int \frac{1}{\sqrt{1 + (ax+b)^2}} \, dx$

Accepted Answer

$\frac{1}{a} \sinh^{-1}(ax+b)$ + C

Question 23

$\int \frac{1}{\sqrt{(ax+b)^2 - 1}} \, dx$

Accepted Answer

$\frac{1}{a} \cosh^{-1}(ax+b)$ + C

Question 24

$\int \frac{1}{1 - (ax+b)^2} \, dx$

Accepted Answer

$\frac{1}{a} 	anh^{-1}(ax+b)$ + C

Integration

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