Integration formulas

45 standard integrals, each with its proof. Search for the one you need.

Updated 28 October 2026Free · no sign-up

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Integral	Result
∫ sin x dx	−cos x + C
∫ cos x dx	sin x + C
∫ tan x dx	log\|sec x\| + C
∫ cot x dx	log\|sin x\| + C
∫ sec x dx	log\|sec x + tan x\| + C
∫ cosec x dx	log\|cosec x − cot 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
∫ log x dx	x log x − x + C
∫ eˣ dx	eˣ + C
∫ aˣ dx	aˣ / log a + C
∫ 1/x dx	log\|x\| + C
∫ xⁿ dx	xⁿ⁺¹/(n + 1) + C, n ≠ −1
∫ sin²x dx	x/2 − (sin 2x)/4 + C
∫ cos²x dx	x/2 + (sin 2x)/4 + C
∫ tan²x dx	tan x − x + C
∫ sin³x dx	cos³x/3 − cos x + C
∫ sin x cos x dx	(sin²x)/2 + C
∫ sec³x dx	½(sec x tan x + log\|sec x + tan x\|) + C
∫ 1/(1 + x²) dx	tan⁻¹x + C
∫ 1/√(1 − x²) dx	sin⁻¹x + C
∫ 1/(x² + a²) dx	(1/a) tan⁻¹(x/a) + C
∫ 1/(x² − a²) dx	(1/2a) log\|(x − a)/(x + a)\| + C
∫ 1/(a² − x²) dx	(1/2a) log\|(a + x)/(a − x)\| + C
∫ 1/√(a² − x²) dx	sin⁻¹(x/a) + C
∫ 1/√(x² + a²) dx	log\|x + √(x² + a²)\| + C
∫ 1/√(x² − a²) dx	log\|x + √(x² − a²)\| + C
∫ √(a² − x²) dx	(x/2)√(a² − x²) + (a²/2) sin⁻¹(x/a) + C
∫ √(x² + a²) dx	(x/2)√(x² + a²) + (a²/2) log\|x + √(x² + a²)\| + C
∫ √(x² − a²) dx	(x/2)√(x² − a²) − (a²/2) log\|x + √(x² − a²)\| + C
∫ x eˣ dx	(x − 1)eˣ + C
∫ x sin x dx	sin x − x cos x + C
∫ x cos x dx	x sin x + cos x + C
∫ x log x dx	(x²/2) log x − x²/4 + C
∫ eˣ sin x dx	(eˣ/2)(sin x − cos x) + C
∫ eˣ cos x dx	(eˣ/2)(sin x + cos x) + C
∫ sin⁻¹x dx	x sin⁻¹x + √(1 − x²) + C
∫ tan⁻¹x dx	x tan⁻¹x − ½ log(1 + x²) + C
∫ 1/(1 + cos x) dx	tan(x/2) + C
∫ 1/(1 + sin x) dx	tan x − sec x + C
∫ (log x)/x dx	(log x)²/2 + C
∫ 1/(x log x) dx	log\|log x\| + C
∫ eˣ[f(x) + f′(x)] dx	eˣ f(x) + C

How to pick a method

Substitution when you can see a function and its derivative: ∫ (log x)/x dx, put t = log x.
By parts for a product of two different kinds of function. Choose u by ILATE: Inverse trig, Log, Algebraic, Trig, Exponential.
Partial fractions for a ratio of polynomials, like 1/(x² − a²).
Trig identities for powers: sin²x = (1 − cos 2x)/2, tan²x = sec²x − 1.

Log means natural log here (base e), the way NCERT and JEE write it. Add the constant C to every indefinite integral.

Questions students ask

What is the integration of 1/x?

log|x| + C.

What is ILATE?

An order for choosing u in integration by parts: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential. Pick u from whichever comes first.

What is the integration of sec x?

log|sec x + tan x| + C.

What is the integration of log x?

x log x − x + C, by parts.

Integration formulas

How to pick a method

Questions students ask

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