This page provides a reference for integrals that frequently appear in upper-division physics courses,
including quantum mechanics, electromagnetism, statistical mechanics, and classical mechanics.
Disclaimer: Always verify the integrals from another source before using them.
Exponential and Gaussian Integrals
Gaussian Integrals
\begin{aligned}
\int_{-\infty}^{\infty} e^{-ax^2} \, dx &= \sqrt{\frac{\pi}{a}}, \quad a > 0 \\
\int_{-\infty}^{\infty} x^2 e^{-ax^2} \, dx &= \frac{1}{2}\sqrt{\frac{\pi}{a^3}}, \quad a > 0 \\
\int_{-\infty}^{\infty} x^4 e^{-ax^2} \, dx &= \frac{3}{4}\sqrt{\frac{\pi}{a^5}}, \quad a > 0 \\
\int_0^{\infty} e^{-ax^2} \, dx &= \frac{1}{2}\sqrt{\frac{\pi}{a}}, \quad a > 0
\end{aligned}
These integrals are fundamental in quantum mechanics (wave function normalization), statistical mechanics
(partition functions), and probability theory. The general form for even powers is:
$$\int_{-\infty}^{\infty} x^{2n} e^{-ax^2} \, dx = \frac{(2n-1)!!}{2^{n+1}}\sqrt{\frac{\pi}{a^{2n+1}}}, \quad a > 0$$
where $(2n-1)!! = (2n-1)(2n-3)\cdots(3)(1)$ is the double factorial.
Example: Normalization of a Gaussian Wave Function
In quantum mechanics, a Gaussian wave packet is given by $\psi(x) = A e^{-x^2/(2\sigma^2)}$. To find the
normalization constant $A$, we require:
$$\int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = |A|^2 \int_{-\infty}^{\infty} e^{-x^2/\sigma^2} \, dx = 1$$
Using the Gaussian integral with $a = 1/\sigma^2$:
$$|A|^2 \sqrt{\pi\sigma^2} = 1 \quad \Rightarrow \quad |A| = \frac{1}{(\pi\sigma^2)^{1/4}}$$
Exponential Integrals
\begin{aligned}
\int e^{ax} \, dx &= \frac{1}{a}e^{ax} + C, \quad a \neq 0 \\
\int x e^{ax} \, dx &= \frac{e^{ax}}{a^2}(ax - 1) + C \\
\int x^2 e^{ax} \, dx &= \frac{e^{ax}}{a^3}(a^2x^2 - 2ax + 2) + C \\
\int_0^{\infty} e^{-ax} \, dx &= \frac{1}{a}, \quad a > 0 \\
\int_0^{\infty} x^n e^{-ax} \, dx &= \frac{n!}{a^{n+1}}, \quad a > 0, \, n \in \mathbb{N}
\end{aligned}
Trigonometric Integrals
Basic Trigonometric Integrals
\begin{aligned}
\int \sin(ax) \, dx &= -\frac{1}{a}\cos(ax) + C \\
\int \cos(ax) \, dx &= \frac{1}{a}\sin(ax) + C \\
\int \tan(ax) \, dx &= -\frac{1}{a}\ln|\cos(ax)| + C \\
\int \sec^2(ax) \, dx &= \frac{1}{a}\tan(ax) + C \\
\int \csc^2(ax) \, dx &= -\frac{1}{a}\cot(ax) + C
\end{aligned}
Powers of Trigonometric Functions
\begin{aligned}
\int \sin^2(ax) \, dx &= \frac{x}{2} - \frac{\sin(2ax)}{4a} + C \\
\int \cos^2(ax) \, dx &= \frac{x}{2} + \frac{\sin(2ax)}{4a} + C \\
\int \sin(ax)\cos(ax) \, dx &= \frac{1}{2a}\sin^2(ax) + C = -\frac{1}{2a}\cos^2(ax) + C \\
\int \sin^n(ax)\cos(ax) \, dx &= \frac{1}{a(n+1)}\sin^{n+1}(ax) + C, \quad n \neq -1 \\
\int \cos^n(ax)\sin(ax) \, dx &= -\frac{1}{a(n+1)}\cos^{n+1}(ax) + C, \quad n \neq -1
\end{aligned}
Example: Average Value in Quantum Mechanics
The expectation value of position for a particle in a box (with wave function $\psi_n(x) = \sqrt{2/L}\sin(n\pi x/L)$)
requires evaluating:
$$\langle x \rangle = \int_0^L x \sin^2\left(\frac{n\pi x}{L}\right) \, dx$$
Using the identity $\sin^2(\theta) = \frac{1}{2}(1 - \cos(2\theta))$ and the integral formulas above,
one finds $\langle x \rangle = L/2$ for all $n$, as expected from symmetry.
Rational Functions
Rational Function Integrals
\begin{aligned}
\int \frac{1}{x^2 + a^2} \, dx &= \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C \\
\int \frac{x}{x^2 + a^2} \, dx &= \frac{1}{2}\ln(x^2 + a^2) + C \\
\int \frac{1}{x^2 - a^2} \, dx &= \frac{1}{2a}\ln\left|\frac{x - a}{x + a}\right| + C \\
\int \frac{1}{\sqrt{a^2 - x^2}} \, dx &= \arcsin\left(\frac{x}{a}\right) + C, \quad |x| < |a| \\
\int \frac{1}{\sqrt{x^2 \pm a^2}} \, dx &= \ln\left|x + \sqrt{x^2 \pm a^2}\right| + C
\end{aligned}
Example: Electric Field from a Line Charge
The electric field at a distance $r$ from an infinite line charge with linear charge density $\lambda$
requires evaluating an integral that reduces to:
$$E = \frac{\lambda}{2\pi\epsilon_0 r} = \frac{\lambda}{2\pi\epsilon_0} \int \frac{dx}{x^2 + r^2}$$
The integral $\int dx/(x^2 + r^2)$ appears frequently in electrostatics problems involving cylindrical symmetry.
Integrals with Square Roots
Square Root Integrals
\begin{aligned}
\int \sqrt{a^2 - x^2} \, dx &= \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\arcsin\left(\frac{x}{a}\right) + C \\
\int \sqrt{x^2 \pm a^2} \, dx &= \frac{x}{2}\sqrt{x^2 \pm a^2} \pm \frac{a^2}{2}\ln\left|x + \sqrt{x^2 \pm a^2}\right| + C \\
\int \frac{x}{\sqrt{a^2 - x^2}} \, dx &= -\sqrt{a^2 - x^2} + C \\
\int \frac{x}{\sqrt{x^2 \pm a^2}} \, dx &= \sqrt{x^2 \pm a^2} + C
\end{aligned}
Logarithmic and Inverse Trigonometric Integrals
Logarithmic Integrals
\begin{aligned}
\int \ln(x) \, dx &= x\ln(x) - x + C \\
\int x^n \ln(x) \, dx &= \frac{x^{n+1}}{n+1}\left(\ln(x) - \frac{1}{n+1}\right) + C, \quad n \neq -1 \\
\int \frac{\ln(x)}{x} \, dx &= \frac{1}{2}[\ln(x)]^2 + C \\
\int \frac{1}{x\ln(x)} \, dx &= \ln|\ln(x)| + C
\end{aligned}
Inverse Trigonometric Integrals
\begin{aligned}
\int \arcsin(x) \, dx &= x\arcsin(x) + \sqrt{1 - x^2} + C \\
\int \arccos(x) \, dx &= x\arccos(x) - \sqrt{1 - x^2} + C \\
\int \arctan(x) \, dx &= x\arctan(x) - \frac{1}{2}\ln(1 + x^2) + C \\
\int x\arctan(x) \, dx &= \frac{1}{2}(x^2 + 1)\arctan(x) - \frac{x}{2} + C
\end{aligned}
Definite Integrals
Useful Definite Integrals
\begin{aligned}
\int_0^{\pi} \sin(nx)\sin(mx) \, dx &= \begin{cases} 0 & \text{if } n \neq m \\ \pi/2 & \text{if } n = m \neq 0 \end{cases} \\
\int_0^{\pi} \cos(nx)\cos(mx) \, dx &= \begin{cases} 0 & \text{if } n \neq m \\ \pi/2 & \text{if } n = m \neq 0 \\ \pi & \text{if } n = m = 0 \end{cases} \\
\int_0^{\pi} \sin(nx)\cos(mx) \, dx &= 0 \quad \text{for all } n, m \\
\int_0^{2\pi} e^{inx} e^{-imx} \, dx &= \begin{cases} 0 & \text{if } n \neq m \\ 2\pi & \text{if } n = m \end{cases} \\
\int_0^{\infty} \frac{\sin(ax)}{x} \, dx &= \begin{cases} \pi/2 & \text{if } a > 0 \\ 0 & \text{if } a = 0 \\ -\pi/2 & \text{if } a < 0 \end{cases}
\end{aligned}
These orthogonality relations are fundamental in Fourier series, quantum mechanics (orthonormal bases),
and signal processing.
Example: Fourier Series Coefficients
When expanding a function $f(x)$ in a Fourier series on $[-\pi, \pi]$:
$$f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} [a_n\cos(nx) + b_n\sin(nx)]$$
The orthogonality relations allow us to compute the coefficients:
$$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos(nx) \, dx, \quad b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin(nx) \, dx$$
The orthogonality ensures that only the $n$-th term contributes when computing $a_n$ or $b_n$.
Special Functions and Asymptotic Forms
Integrals Leading to Special Functions
\begin{aligned}
\int_0^{\infty} x^{n-1} e^{-x} \, dx &= \Gamma(n), \quad n > 0 \quad \text{(Gamma function)} \\
\int_0^1 x^{m-1}(1-x)^{n-1} \, dx &= \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)} = B(m,n) \quad \text{(Beta function)} \\
\int_0^{\infty} \frac{x^{s-1}}{e^x - 1} \, dx &= \Gamma(s)\zeta(s), \quad s > 1 \quad \text{(Riemann zeta function)}
\end{aligned}
The Gamma function $\Gamma(n) = (n-1)!$ for positive integers and appears in statistical mechanics
(partition functions) and quantum field theory.
Summary
- Gaussian integrals: Essential for quantum mechanics and statistical mechanics; $\int_{-\infty}^{\infty} e^{-ax^2} \, dx = \sqrt{\pi/a}$
- Trigonometric integrals: Frequently appear in wave mechanics, oscillations, and Fourier analysis
- Orthogonality relations: Fundamental for Fourier series, quantum mechanics bases, and signal processing
- Rational functions: Common in electrostatics and potential theory
- Special functions: Gamma, Beta, and zeta functions appear in advanced statistical mechanics and field theory
- Many of these integrals can be derived using the techniques covered in the Integration Techniques page