Integration Rules & Formulas
A complete reference of all essential integration rules and formulas. Master these rules to solve any integral you encounter.
Basic Integration Rules
These fundamental rules form the foundation of all integration problems.
Constant Rule
$$\int k \, dx = kx + C$$
The integral of a constant k is kx plus C.
Constant Multiple Rule
$$\int k \cdot f(x) \, dx = k \int f(x) \, dx$$
Constants can be factored out of integrals.
Sum/Difference Rule
$$\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$$
The integral of a sum equals the sum of integrals.
Power Rule
The most frequently used integration rule.
Power Rule
$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$
Special Case: n = -1
$$\int \frac{1}{x} \, dx = \int x^{-1} \, dx = \ln|x| + C$$
Examples
- ∫ x³ dx = x⁴/4 + C
- ∫ x⁻² dx = -x⁻¹ + C = -1/x + C
- ∫ √x dx = ∫ x^(1/2) dx = (2/3)x^(3/2) + C
Exponential Rules
Natural Exponential
$$\int e^x \, dx = e^x + C$$
General Exponential
$$\int a^x \, dx = \frac{a^x}{\ln a} + C, \quad a > 0, a \neq 1$$
Exponential with Linear Argument
$$\int e^{ax} \, dx = \frac{e^{ax}}{a} + C$$
Logarithmic Rules
Natural Logarithm
$$\int \ln x \, dx = x \ln x - x + C$$
Logarithm Base a
$$\int \log_a x \, dx = \frac{x \ln x - x}{\ln a} + C$$
Trigonometric Rules
Essential formulas for integrating trigonometric functions.
Sine
$$\int \sin x \, dx = -\cos x + C$$
Cosine
$$\int \cos x \, dx = \sin x + C$$
Tangent
$$\int \tan x \, dx = -\ln|\cos x| + C$$
Cotangent
$$\int \cot x \, dx = \ln|\sin x| + C$$
Secant
$$\int \sec x \, dx = \ln|\sec x + \tan x| + C$$
Cosecant
$$\int \csc x \, dx = -\ln|\csc x + \cot x| + C$$
Secant Squared
$$\int \sec^2 x \, dx = \tan x + C$$
Cosecant Squared
$$\int \csc^2 x \, dx = -\cot x + C$$
Secant Tangent
$$\int \sec x \tan x \, dx = \sec x + C$$
Cosecant Cotangent
$$\int \csc x \cot x \, dx = -\csc x + C$$
Inverse Trigonometric Rules
Arcsine Form
$$\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + C$$
Arctangent Form
$$\int \frac{1}{1+x^2} \, dx = \arctan x + C$$
Arcsecant Form
$$\int \frac{1}{x\sqrt{x^2-1}} \, dx = \text{arcsec}|x| + C$$
General Arcsine
$$\int \frac{1}{\sqrt{a^2-x^2}} \, dx = \arcsin\frac{x}{a} + C$$
General Arctangent
$$\int \frac{1}{a^2+x^2} \, dx = \frac{1}{a}\arctan\frac{x}{a} + C$$
Hyperbolic Function Rules
Hyperbolic Sine
$$\int \sinh x \, dx = \cosh x + C$$
Hyperbolic Cosine
$$\int \cosh x \, dx = \sinh x + C$$
Hyperbolic Tangent
$$\int \tanh x \, dx = \ln(\cosh x) + C$$
Hyperbolic Secant²
$$\int \text{sech}^2 x \, dx = \tanh x + C$$
Special Integrals
Square Root of Sum of Squares
$$\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2-x^2} +
\frac{a^2}{2}\arcsin\frac{x}{a} + C$$
Rational with Quadratic Denominator
$$\int \frac{1}{x^2 - a^2} \, dx =
\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$$
Tips for Using These Rules
- Always add + C for indefinite integrals
- Simplify first before integrating when possible
- Factor out constants to make integration easier
- Rewrite expressions using equivalent forms that match known rules
- Verify your answer by differentiating it