Integral Rules

Basic Integration Rules

Rule	Formula
Constant Rule	∫ c dx = cx + C
Power Rule	∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)
∫ x⁻¹ dx	ln\|x\| + C
Sum Rule	∫ [f(x) ± g(x)] dx = ∫f dx ± ∫g dx
Constant Multiple	∫ c·f(x) dx = c·∫f(x) dx
Integration by Parts	∫ u dv = uv - ∫ v du
u-Substitution	∫ f(g(x))·g'(x) dx = ∫ f(u) du, u=g(x)

Common Integrals

f(x)	∫f(x) dx
eˣ	eˣ + C
aˣ	aˣ / ln(a) + C
sin(x)	-cos(x) + C
cos(x)	sin(x) + C
tan(x)	-ln\|cos(x)\| + C
sec²(x)	tan(x) + C
csc²(x)	-cot(x) + C
sec(x)tan(x)	sec(x) + C
1/√(1-x²)	arcsin(x) + C
1/(1+x²)	arctan(x) + C
1/√(x²+a²)	ln(x + √(x²+a²)) + C
√(a²-x²)	(x/2)√(a²-x²) + (a²/2)arcsin(x/a) + C