Reduction Formulas for Integration Cheat Sheet

Key Facts

• A reduction formula rewrites an integral sequence $I_n$ in terms of a simpler integral such as $I_{n-1}$ or $I_{n-2}$.
• For $I_n=\int \sin^n x\,dx$, the reduction formula is $I_n=-\frac{\sin^{n-1}x\cos x}{n}+\frac{n-1}{n}I_{n-2}$ for $n\ge 2$.
• For $I_n=\int \cos^n x\,dx$, the reduction formula is $I_n=\frac{\cos^{n-1}x\sin x}{n}+\frac{n-1}{n}I_{n-2}$ for $n\ge 2$.
• For $I_n=\int \tan^n x\,dx$, the reduction formula is $I_n=\frac{\tan^{n-1}x}{n-1}-I_{n-2}$ for $n\ne 1$ and $n\ge 2$.
• For $I_n=\int \sec^n x\,dx$, the reduction formula is $I_n=\frac{\sec^{n-2}x\tan x}{n-1}+\frac{n-2}{n-1}I_{n-2}$ for $n>1$.
• For $I_n=\int x^n e^{ax}\,dx$, integration by parts gives $I_n=\frac{x^n e^{ax}}{a}-\frac{n}{a}I_{n-1}$ when $a\ne 0$.
• For $I_n=\int x^n\ln x\,dx$, integration by parts gives $I_n=\frac{x^{n+1}\ln x}{n+1}-\frac{x^{n+1}}{(n+1)^2}+C$ when $n\ne -1$.
• A reduction process must stop at a base case such as $I_0=\int 1\,dx=x+C$, $I_1=\int \sin x\,dx=-\cos x+C$, or $I_1=\int \sec x\,dx=\ln|\sec x+\tan x|+C$.

Vocabulary

Reduction formula

A formula that expresses an integral with parameter $n$ in terms of a related integral with a smaller parameter.

Base case

The simplest integral in a reduction chain, such as $I_0$ or $I_1$, that can be evaluated directly.

Integration by parts

A method based on $\int u\,dv=uv-\int v\,du$ that is often used to derive reduction formulas.

Recursive relation

An equation that defines one term, such as $I_n$, using earlier terms such as $I_{n-1}$ or $I_{n-2}$.

Trigonometric power integral

An integral involving powers of trigonometric functions, such as $\int \sin^n x\,dx$ or $\int \sec^n x\,dx$.

Parameter

A symbol such as $n$ or $a$ that represents a fixed value while the integration variable, usually $x$, changes.