The table of integrals and the simplest rules

Table: ∫ xᵖ dx = xᵖ⁺¹/(p + 1) + C (p ≠ −1); ∫ dx/x = ln|x| + C; ∫ eˣ dx = eˣ + C; ∫ aˣ dx = aˣ/ln a + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C; ∫ dx/cos²x = tan x + C; ∫ dx/sin²x = −cot x + C. Linear argument: if ∫ f(x) dx = F(x) + C then ∫ f(kx + b) dx = (1/k)F(kx + b) + C (k ≠ 0); for example ∫ cos(kx + b) dx = (1/k) sin(kx + b) + C. Rules: ∫ a·f dx = a ∫ f dx; ∫ (f ± g) dx = ∫ f dx ± ∫ g dx. The integral of a product is not the product of the integrals. For more complicated integrals substitute a new variable: for example u = x² + 5 gives du = 2x dx. If needed, simplify the integrand with trigonometric identities (cos²x = (1 + cos 2x)/2 and others). Always check the answer by differentiating.