Basic Standard Integrals:

• integral x^n dx = x^(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 1/(1+x^2) dx = arctan(x) + C
• integral 1/(x*sqrt(x^2-1)) dx = arcsec(x) + C

Standard Forms with Parameters:

• integral 1/(x^2+a^2) dx = (1/a)arctan(x/a) + C
• integral 1/(x^2-a^2) dx = (1/2a)ln|(x-a)/(x+a)| + C
• integral 1/(a^2-x^2) dx = (1/2a)ln|(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(x/a) + C
• integral sqrt(a^2-x^2) dx = (x/2)sqrt(a^2-x^2) + (a^2/2)arcsin(x/a) + C
• integral sqrt(x^2+a^2) dx = (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(x/2)):

• sin x = 2t/(1+t^2), cos x = (1-t^2)/(1+t^2), dx = 2dt/(1+t^2)

Reduction Formulas:

• I_n(sin) = -(1/n)sin^(n-1)x cos x + ((n-1)/n)I_(n-2)
• I_n(cos) = (1/n)cos^(n-1)x sin x + ((n-1)/n)I_(n-2)
• I_n(tan) + I_(n-2)(tan) = tan^(n-1)x/(n-1)