Integration Rules: Table of Integrals and Techniques

Q: What are the basic integration rules?

Power rule \int x^n dx = \frac{x^{n+1}}{n+1} + C (n \ne -1), \int \frac1x dx = \ln|x| + C, \int e^x dx = e^x + C, \int \sin x\,dx = -\cos x + C, \int \cos x\,dx = \sin x + C, plus linearity: constants come out, and sums split.

Q: What is LIATE?

An order for choosing u in integration by parts: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Pick the one that comes first as u.

Why “+ C”?

Integration undoes differentiation. But differentiation throws away constants, since the derivative of 5 is 0, so undoing it can’t know which constant was there. Every antiderivative is really a whole family of curves stacked vertically:

C = 0.0

Every curve $F(x) = \tfrac{x^3}{3} - x + C$ has the same derivative, $x^2 - 1$ . Drag across the graph: at any x, all the tangent lines are parallel. Sliding C moves the curve up and down without changing a single slope, so the derivative can’t tell them apart, and the integral must include $+\,C$ .

Linearity

$\int \big(a f(x) + b g(x)\big)\,dx = a\int f(x)\,dx + b\int g(x)\,dx$ . Constants come out and sums split, so polynomials integrate term by term.

Table of basic integrals

Integral	Result
$\int k\,dx$	$kx + C$
$\int x^n\,dx$	$\frac{x^{n+1}}{n+1} + C\quad (n \ne -1)$
$\int \frac{1}{x}\,dx$	$\ln\|x\| + C$
$\int e^x\,dx$	$e^x + C$
$\int a^x\,dx$	$\frac{a^x}{\ln a} + C$
$\int \ln x\,dx$	$x\ln x - x + C$
$\int \sin x\,dx$	$-\cos x + C$
$\int \cos x\,dx$	$\sin x + C$
$\int \sec^2 x\,dx$	$\tan x + C$
$\int \tan x\,dx$	$-\ln\|\cos x\| + C$
$\int \sec x\,dx$	$\ln\|\sec x + \tan x\| + C$
$\int \frac{1}{1+x^2}\,dx$	$\arctan x + C$
$\int \frac{1}{\sqrt{1-x^2}}\,dx$	$\arcsin x + C$

Techniques

u-substitution (the chain rule in reverse)

Look for a function and its derivative. In $\int 2x\cos(x^2)\,dx$ , let $u = x^2$ , $du = 2x\,dx$ : the integral becomes $\int \cos u\,du = \sin u + C = \sin(x^2) + C$ .

Integration by parts (the product rule in reverse)

$\int u\,dv = uv - \int v\,du$ . For $\int x e^x\,dx$ take $u = x$ , $dv = e^x dx$ : result $xe^x - e^x + C$ . Choose $u$ by LIATE.

Partial fractions

Split a rational function into simple pieces: $\frac{1}{x^2-1} = \frac{1/2}{x-1} - \frac{1/2}{x+1}$ , each of which integrates to a logarithm.

Trig identities

Reduce powers first: $\sin^2 x = \frac{1 - \cos 2x}{2}$ , so $\int \sin^2 x\,dx = \frac{x}{2} - \frac{\sin 2x}{4} + C$ .

The integral calculator picks the technique automatically and shows each step. Single results with explanations: ∫ ln x, ∫ tan x, ∫ sec x, all integrals.

Questions students ask

What are the basic integration rules?

Power rule $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ ( $n \ne -1$ ), $\int \frac1x dx = \ln|x| + C$ , $\int e^x dx = e^x + C$ , $\int \sin x\,dx = -\cos x + C$ , $\int \cos x\,dx = \sin x + C$ , plus linearity: constants come out, and sums split.

Why is there no product rule for integrals?

The integral of a product is not the product of integrals. Instead, the product rule for derivatives turns into integration by parts: $\int u\,dv = uv - \int v\,du$ .

How do I choose between substitution and parts?

If one factor is (a multiple of) the derivative of something inside another factor, use substitution. If you have a product of unrelated types (polynomial × exponential, polynomial × trig, log × anything), use parts, choosing $u$ with LIATE.

What is LIATE?

An order for choosing $u$ in integration by parts: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential. Pick the one that comes first as $u$ .