Basic Standard Integrals:

• integral $x^n$ dx = x^$\frac{n+1}{(n+1)}$ + C, n != -1
• integral 1/x dx = ln|x| + C
• integral $e^x$ dx = $e^x$ + C
• integral $a^x$ dx = $a^x$/ln(a) + C
• integral sin x dx = -cos x + C
• integral cos x dx = sin x + C
• integral $sec^2$ x dx = tan x + C
• integral $csc^2$ x dx = -cot x + C
• integral sec x tan x dx = sec x + C
• integral csc x cot x dx = -csc x + C

Inverse Trigonometric Integrals:

• integral 1/sqrt(1-$x^2$) dx = arcsin(x) + C
• integral $\frac{1}{1+x^2}$ dx = arctan(x) + C
• integral $\frac{1}{x*sqrt(x^2-1}$) dx = arcsec(x) + C

Standard Forms with Parameters:

• integral $\frac{1}{x^2+a^2}$ dx = $\frac{1}{a}$arctan$\frac{x}{a}$ + C
• integral $\frac{1}{x^2-a^2}$ dx = $\frac{1}{2a}$ln|$\frac{x-a}{(x+a)}$| + C
• integral $\frac{1}{a^2-x^2}$ dx = $\frac{1}{2a}$ln|$\frac{a+x}{(a-x)}$| + C
• integral 1/sqrt(x^{2+a}^2) dx = ln|x+sqrt(x^{2+a}^2)| + C
• integral 1/sqrt(x^{2-a}^2) dx = ln|x+sqrt(x^{2-a}^2)| + C
• integral 1/sqrt(a^{2-x}^2) dx = arcsin$\frac{x}{a}$ + C
• integral sqrt(a^{2-x}^2) dx = $\frac{x}{2}$sqrt(a^{2-x}^2) + ($a^2$/2)arcsin$\frac{x}{a}$ + C
• integral sqrt(x^{2+a}^2) dx = $\frac{x}{2}$sqrt(x^{2+a}^2) + ($a^2$/2)ln|x+sqrt(x^{2+a}^2)| + C

Integration by Parts: integral u dv = uv - integral v du (LIATE order for u)

Special Forms:

• integral $e^x$[f(x)+f'(x)] dx = $e^x$*f(x) + C
• integral sec x dx = ln|sec x + tan x| + C
• integral csc x dx = ln|csc x - cot x| + C

Weierstrass Substitution (t = tan$\frac{x}{2}$):

• sin x = 2$\frac{t}{1+t^2}$, cos x = $\frac{1-t^2}{(1+t^2)}$, dx = 2$\frac{dt}{1+t^2}$

Reduction Formulas:

• $I_n$(sin) = -$\frac{1}{n}$sin^(n-1)x cos x + $\frac{(n-1}{n}$)I_(n-2)
• $I_n$(cos) = $\frac{1}{n}$cos^(n-1)x sin x + $\frac{(n-1}{n}$)I_(n-2)
• $I_n$(tan) + I_(n-2)(tan) = tan^(n-1)$\frac{x}{n-1}$