Integral Rules

Fundamental theorem of calculus

Part 1

\frac{d}{d x} \int_{a}^{g (x)} f (t) d t = f (g (x)) \cdot g^{'} (x)

Part 2

\int_{a}^{b} f (x) d x = F (b) - F (a)

where F′ = f

Power

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

n ≠ −1

One over x

\int \frac{1}{x} d x = \ln | x | + C

e to the x

\int e^{x} d x = e^{x} + C

a to the x

\int a^{x} d x = \frac{a^{x}}{\ln a} + C

Sine

\int \sin x d x = - \cos x + C

Cosine

\int \cos x d x = \sin x + C

Secant squared

\int \sec^{2} x d x = \tan x + C

Cosecant squared

\int \csc^{2} x d x = - \cot x + C

Secant times tangent

\int \sec x \tan x d x = \sec x + C

Cosecant times cotangent

\int \csc x \cot x d x = - \csc x + C

Tangent

\int \tan x d x = - \ln | \cos x | + C = \ln | \sec x | + C

Secant

\int \sec x d x = \ln | \sec x + \tan x | + C

Inverse sine form

\int \frac{1}{\sqrt{a^{2} - x^{2}}} d x = \arcsin \frac{x}{a} + C

Inverse tangent form

\int \frac{1}{a^{2} + x^{2}} d x = \frac{1}{a} \arctan \frac{x}{a} + C

Integration by parts

\int u d v = u v - \int v d u

Trapezoidal rule

\int_{a}^{b} f (x) d x \approx \frac{h}{2} [f (x_{0}) + 2 f (x_{1}) + \dots + 2 f (x_{n - 1}) + f (x_{n})]

h = (b − a)/n

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