Indefinite integrals

The essentials

If F′(x) = f(x), then F is called an antiderivative (or primitive) of f, and we write ∫ f(x) dx = F(x) + C, where C is an arbitrary constant called the constant of integration. The constant is needed because adding any constant to F does not change its derivative, so a function has infinitely many antiderivatives, all differing by constants. Geometrically, they form a family of parallel curves shifted vertically.

Integration is linear: ∫ [a f(x) + b g(x)] dx = a ∫ f(x) dx + b ∫ g(x) dx. So you can split sums and pull out constants, then use standard forms.

Key points

• ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1.
• ∫ (1/x) dx = log|x| + C.
• ∫ eˣ dx = eˣ + C; ∫ e^(kx) dx = e^(kx)/k + C; ∫ aˣ dx = aˣ/log a + C.
• ∫ cos x dx = sin x + C; ∫ sin x dx = −cos x + C.
• ∫ sec²x dx = tan x + C; ∫ cosec²x dx = −cot x + C.
• ∫ 1/√(1 − x²) dx = sin⁻¹x + C; ∫ 1/(1 + x²) dx = tan⁻¹x + C.
• A given condition such as F(1) = 5 fixes the value of C and picks one curve from the family.

Exam tips

• Check any answer by differentiating it; you should get back the integrand.
• Rewrite roots and fractions as powers first: 1/√x = x^(−1/2).

Common mistakes

• Applying the power rule to 1/x, which gives division by zero; its integral is log|x|.
• Writing ∫ sin x dx = cos x + C (the sign is negative).
• Leaving out + C in an indefinite integral.