Integral Calculator

Calculate definite and indefinite integrals with step-by-step solutions. Supports multiple integration techniques and provides detailed explanations.

Function to Integrate

Use: +, -, *, /, ^, sin(), cos(), tan(), ln(), exp(), sqrt()

Integration Variable

Integration Method

Quick Functions

Function to Integrate

Integration Bounds

Lower Bound (a)

∫

Upper Bound (b)

Integration Variable

Calculation Method

Quick Examples

Integration Technique

U-Substitution

∫ f(g(x))g'(x) dx = ∫ f(u) du

Integration by Parts

∫ u dv = uv - ∫ v du

Partial Fractions

Decompose rational functions

Trigonometric Substitution

For √(a² ± x²) expressions

Function f(x)

Substitution u =

Hint: Choose u so that du appears in the integrand

Function to Integrate

Improper Integral Type

Integration Bounds

Lower Bound

∫

Upper Bound

Quick Examples

Integration Rules & Techniques

Integration is the reverse process of differentiation. It finds the antiderivative of a function and can calculate areas under curves, volumes, and many other applications.

Basic Integration Rules

Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (n ≠ -1)

Exponential: $\int e^x dx = e^x + C$

Logarithmic: $\int \frac{1}{x} dx = \ln|x| + C$

Trigonometric: $\int \sin x dx = -\cos x + C$

Sum Rule: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

Integration Techniques

U-Substitution

Used when the integrand contains a function and its derivative.

$\int f(g(x))g'(x) dx = \int f(u) du$ where $u = g(x)$

Integration by Parts

Used for products of functions, especially with LIATE rule.

$\int u dv = uv - \int v du$

Partial Fractions

Decomposes rational functions into simpler fractions.

$\frac{P(x)}{Q(x)} = \frac{A}{x-a} + \frac{B}{x-b} + ...$

Trigonometric Substitution

For integrands containing √(a² ± x²) expressions.

Use x = a sin θ, x = a tan θ, or x = a sec θ

Common Integrals

Function	Integral
$x^n$	$\frac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln\|x\| + C$
$e^x$	$e^x + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$

Definite vs Indefinite Integrals

Indefinite Integral

Represents a family of functions (antiderivatives).

$\int f(x) dx = F(x) + C$

Definite Integral

Calculates the exact area under a curve between bounds.

$\int_a^b f(x) dx = F(b) - F(a)$

Integration Tip:

Always check your integration result by differentiating it. The derivative of your antiderivative should equal the original function (plus any constant terms).

Integration Examples

Polynomial Integration

∫(x³ + 2x² + x) dx

Result: x⁴/4 + 2x³/3 + x²/2 + C

Integration by Parts

∫x·eˣ dx

Use u = x, dv = eˣ dx

Definite Integral

∫₀^π sin(x) dx

Result: 2

Improper Integral

∫₁^∞ 1/x² dx

Convergent: Result = 1