MATH-2414: Calculus II

$kx+C$

$\frac{1}{n+1}x^{n+1}+C$

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

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

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

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

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

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

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

$\ln|x|+C$

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

$\arcsin(\frac{x}{a})+C$

$\frac{1}{a}arcsec|\frac{x}{a}|+C$

$\frac{1}{a}\ln|\sec(ax)|+C$

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

$\frac{1}{a}\ln|\sec(ax)+\tan(ax)|+C$

$-\frac{1}{a}\ln|\csc(ax)+\cot(ax)|+C$

u-substitution, split the fraction, long division, different of squares, completing the square

$\int u dv = uv -\int vdu$ or $\int uv' = uv -\int vu'$

Differentiate one of the multipliers until you reach zero, integrate the other multiplier. Multiply diagonally switching signs, and sum the results to get the answer.

1. Logarithms
2. Inverses
3. Algebraic (+,-,$\times$,$\div$,$\sqrt()$,$^n$)
4. Trig
5. Exponential

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

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

$cos(x)$

$-sin(x)$

$sec^2 x$

$-csc^2 (x)$

$sec(x)tan(x)$

$-csc(x)cot(x)$

$sin^2 \theta+cos^2\theta=1$
$tan^2 \theta+1=sec^2\theta$
$1+cot^2 \theta=csc^2\theta$

$\int sin(a)sin(b)=\frac{1}{2}\int cos(a-b)-cos(a+b)$
$\int cos(a)cos(b)=\frac{1}{2}\int cos(a-b)+cos(a+b)$
$\int sin(a)cos(b)=\frac{1}{2}\int sin(a-b)+sin(a+b)$

$x=a\sin{\theta}$ simplifies to $a\cos{\theta}$

$x=a\sec{\theta}$ simplifies to $a\tan{\theta}$

$x=a\tan{\theta}$ simplifies to $a\sec{\theta}$

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

$\frac{A}{linear}+\frac{B}{linear}+\frac{C}{quadratic}$

1. Multiple Linear Terms
2. Repeated Linear Terms
3. Quadratic Terms
4. Long Division $(Degree_{Top} \ge Degree_{Bottom})$