Standard integrals

Every derivative formula read backwards gives an integral formula. These are the base cases that every method in the chapter reduces to.

Powers: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C for every n ≠ −1, and ∫dx = x + C. The case n = −1 is separate: ∫(1/x) dx = log|x| + C.

Trigonometric: ∫cos x dx = sin x + C, ∫sin x dx = −cos x + C, ∫sec²x dx = tan x + C, ∫cosec²x dx = −cot x + C, ∫sec x tan x dx = sec x + C, ∫cosec x cot x dx = −cosec x + C.

Inverse trigonometric: ∫dx/√(1 − x²) = sin⁻¹x + C, ∫dx/(1 + x²) = tan⁻¹x + C, ∫dx/(x√(x² − 1)) = sec⁻¹x + C. The same integrands can also be written with −cos⁻¹x, −cot⁻¹x and −cosec⁻¹x; the two answers differ only by a constant.

Exponential: ∫eˣ dx = eˣ + C and ∫aˣ dx = aˣ/log a + C for a > 0, a ≠ 1.

The modulus in log|x| matters: 1/x is defined for negative x too, and log|x| is its antiderivative on both sides of 0.

Check any answer by differentiating it; the result must be the integrand exactly.