Integral Calculator

An integral is a fundamental concept in calculus that represents the accumulation of quantities. There are two main types:

Indefinite Integral (Antiderivative)

The indefinite integral of $f(x)$ is a family of functions $F(x) + C$ such that $F'(x) = f(x)$:

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

where $C$ is the constant of integration.

Definite Integral

The definite integral computes the net signed area under the curve $f(x)$ from $a$ to $b$:

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

This relationship is known as the Fundamental Theorem of Calculus, which connects differentiation and integration.

Geometrically, the definite integral represents the area between the function and the $x$-axis over the interval $[a, b]$. Areas above the axis are positive, and areas below are negative.

Integrals have wide applications in physics (work, displacement), engineering (signal processing), probability (expected values), and economics (consumer surplus).

Basic Integration Rules

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

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

$\int e^x\,dx = e^x + C$

$\int \sin x\,dx = -\cos x + C$

$\int \cos x\,dx = \sin x + C$

Method 1: Substitution (u-substitution)

Used when the integrand contains a composite function. Let $u = g(x)$, then $du = g'(x)\,dx$:

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

Example: $\int 2x \cdot e^{x^2}\,dx$. Let $u = x^2$, $du = 2x\,dx$, so the integral becomes $\int e^u\,du = e^{x^2} + C$.

Method 2: Integration by Parts

Based on the product rule for derivatives:

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

Choose $u$ and $dv$ using the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).

Example: $\int x \cdot e^x\,dx$. Let $u = x$, $dv = e^x\,dx$. Then $du = dx$, $v = e^x$. Result: $xe^x - e^x + C$.

Method 3: Partial Fractions

For rational functions $\frac{P(x)}{Q(x)}$, decompose into simpler fractions:

$\int \frac{1}{x^2 - 1}\,dx = \int \frac{1}{2}\left(\frac{1}{x-1} - \frac{1}{x+1}\right)dx = \frac{1}{2}\ln\left|\frac{x-1}{x+1}\right| + C$

Method 4: Trigonometric Substitution

For integrands involving $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, or $\sqrt{x^2 - a^2}$:

Comparison of Methods

• Forgetting the constant of integration: Every indefinite integral must include $+ C$. The antiderivative is a family of functions.
• Incorrect power rule application: $\int x^{-1}\,dx = \ln|x| + C$, not $\frac{x^0}{0}$. The power rule $\frac{x^{n+1}}{n+1}$ does not apply when $n = -1$.
• Sign errors with trig integrals: $\int \sin x\,dx = -\cos x + C$ (negative sign). $\int \cos x\,dx = \sin x + C$ (positive sign).
• Forgetting to substitute back: When using $u$-substitution, always convert the final answer back to the original variable $x$.
• Wrong bounds in definite integrals: When using substitution in definite integrals, either change the limits to match the new variable or substitute back before evaluating.