Basic Integrals/Derivs/Trig Fns

Question 1

$\int{\frac{du}{u^2-a^2}}$

Accepted Answer

$= \frac{1}{2a}\ln\left|\frac{u - a}{u + a}\right|+C$

Question 2

$\int{\frac{du}{a^2-u^2}}$

Accepted Answer

$= \frac{1}{2a}\ln\left|\frac{u + a}{u - a}\right|+C$

Question 3

$\int\frac{du}{\sqrt{a^2-u^2}}$

Accepted Answer

$= \sin^{-1}\left(\frac{u}{a}\right)+C$

Question 4

$\int\frac{du}{u\sqrt{u^2 - a^2}}$

Accepted Answer

$= \frac{1}{a}\sec^{-1}\left(\frac{u}{a}\right)+C$

Question 5

$\int\frac{du}{a^2 + u^2}$

Accepted Answer

$= \frac{u}{a}	an^{-1}\left(\frac{u}{a}ight)+C$

Question 6

$\sin(x + y)$

Accepted Answer

$= \sin(x)\cos(y)+\cos(x)sin(y)$

Question 7

$\sin(x - y)$

Accepted Answer

$= \sin(x)\cos(y)-\cos(x)\sin(y)$

Question 8

$\cos(x + y)$

Accepted Answer

$= \cos(x)\cos(y)-\sin(x)\sin(y)$

Question 9

$\cos(x - y)$

Accepted Answer

$= \cos(x)\cos(y)+\sin(x)\sin(y)$

Question 10

$	an(x + y)$

Accepted Answer

$= \frac{	an(x) + 	an(y)}{1 - 	an(x)	an(y)}$

Question 11

$	an(x - y)$

Accepted Answer

$= \frac{	an(x) - 	an(y)}{1 + 	an(x)	an(y)}$

Question 12

$\sin(2x)$

Accepted Answer

$= 2\sin(x)\cos(x)$

Question 13

$\cos(2x)$

Accepted Answer

$= \cos^2(x) - \sin^2(x)$ $= 2\cos^2(x) - 1$ $= 1 - 2sin^2(x)$

Question 14

$	an(2x)$

Accepted Answer

$= \frac{2	an(x)}{1 - 	an^2(x)}$

Question 15

$\sin^2(x)$

Accepted Answer

$= \frac{1 - \cos(2x)}{2}$

Question 16

$\cos^2(x)$

Accepted Answer

$= \frac{1 + \cos(2x)}{2}$

Question 17

$\int{b^udu}$

Accepted Answer

$\frac{b^u}{\ln{b}}+C$

Question 18

$\int\sin(u)du$

Accepted Answer

$= -\cos(u)+C$

Question 19

$\int\cos(u)du$

Accepted Answer

$= \sin(u)+C$

Question 20

$\int\sec^2(u)du$

Accepted Answer

$= 	an(u)+C$

Question 21

$\csc^2(u)du$

Accepted Answer

$= -\cot(u)+C$

Question 22

$\int\sec(u)	an(u)du$

Accepted Answer

$= \sec(u)+C$

Question 23

$\int\csc(u)\cot(u)du$

Accepted Answer

$= -\csc(u)+C$

Question 24

$\int	an(u)du$

Accepted Answer

$= \ln|\sec(u)|+C$

Question 25

$\int\cot(u)du$

Accepted Answer

$= \ln|\sin(u)|+C$

Question 26

$\int\sec(u)du$

Accepted Answer

$= \ln|\sec(u)+tan(u)|+C$

Question 27

$\int\csc(u)du$

Accepted Answer

$= \ln|\csc(u)-\cot(u)|+C$

Question 28

$\frac{d}{dx}(\arcsin(x))$

Accepted Answer

$= \frac{1}{\sqrt{1-x^2}}$

Question 29

$\frac{d}{dx}(\arccos(x))$

Accepted Answer

$= -\frac{1}{\sqrt{1-x^2}}$

Question 30

$\frac{d}{dx}(\arctan(x))$

Accepted Answer

$= \frac{1}{1+x^2}$

Question 31

$\frac{d}{dx}(\csc^{-1}(x))$

Accepted Answer

$= -\frac{1}{x\sqrt{x^2-1}}$

Question 32

$\frac{d}{dx}(\sec^{-1}(x))$

Accepted Answer

$= \frac{1}{x\sqrt{x^2-1}}$

Question 33

$\frac{d}{dx}(\cot^{-1}(x))$

Accepted Answer

$= -\frac{1}{1+x^2}$

Basic Integrals/Derivs/Trig Fns

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