Integration using Reduction Formulae

What are Reduction Formulae?

A reduction formula is a mathematical equation that expresses an integral involving a parameter (often an integer power, $n$) in terms of a similar integral but with a reduced value of that parameter (e.g., $n-1$ or $n-2$). Think of it as a recursive recipe for solving complex integrals.

The primary method used to derive these formulae is integration by parts. By strategically choosing $u$ and $dv$, one application of integration by parts can relate an integral $I_n$ to an integral $I_{n-k}$ (where $k$ is usually 1 or 2) and some other terms that do not involve an integral.

The main advantage is that by repeatedly applying the reduction formula, we can systematically simplify the integral until it reaches a "base case"—an integral that is simple enough to be evaluated directly (e.g., $\int \sin x \, dx$ or $\int e^x \, dx$). This process is particularly powerful for integrals of functions like $\sin^n x$, $x^n e^{ax}$, or $(\ln x)^n$, where direct integration for larger $n$ would involve many tedious and error-prone repetitions of integration by parts.

Common Reduction Formulae

Below are some frequently encountered reduction formulae. In these examples, $n$ is typically a positive integer, and $a$ is a constant. The variable of integration is $x$.

• For $\int x^n e^{ax} dx$: $I_n = \frac{1}{a} x^n e^{ax} - \frac{n}{a} \int x^{n-1} e^{ax} dx = \frac{1}{a} x^n e^{ax} - \frac{n}{a} I_{n-1}$
• For $\int x^n \cos(ax) dx$: $I_n = \frac{1}{a} x^n \sin(ax) - \frac{n}{a} \int x^{n-1} \sin(ax) dx$
• For $\int x^n \sin(ax) dx$: $I_n = -\frac{1}{a} x^n \cos(ax) + \frac{n}{a} \int x^{n-1} \cos(ax) dx$
• For $\int \sin^n(x) dx$ ($n \ge 2$): $I_n = -\frac{1}{n} \sin^{n-1}(x)\cos(x) + \frac{n-1}{n} \int \sin^{n-2}(x) dx = -\frac{1}{n} \sin^{n-1}(x)\cos(x) + \frac{n-1}{n} I_{n-2}$
• For $\int \cos^n(x) dx$ ($n \ge 2$): $I_n = \frac{1}{n} \cos^{n-1}(x)\sin(x) + \frac{n-1}{n} \int \cos^{n-2}(x) dx = \frac{1}{n} \cos^{n-1}(x)\sin(x) + \frac{n-1}{n} I_{n-2}$
• For $\int \tan^n(x) dx$ ($n \ge 2$): $I_n = \frac{1}{n-1} \tan^{n-1}(x) - \int \tan^{n-2}(x) dx = \frac{1}{n-1} \tan^{n-1}(x) - I_{n-2}$
• For $\int \sec^n(x) dx$ ($n \ge 2$): $I_n = \frac{1}{n-1} \sec^{n-2}(x)\tan(x) + \frac{n-2}{n-1} \int \sec^{n-2}(x) dx = \frac{1}{n-1} \sec^{n-2}(x)\tan(x) + \frac{n-2}{n-1} I_{n-2}$
• For $\int (\ln x)^n dx$: $I_n = x(\ln x)^n - n \int (\ln x)^{n-1} dx = x(\ln x)^n - n I_{n-1}$

The base cases for these formulae are typically when $n=0$ or $n=1$ (or sometimes $n=2$ for secant/tangent if $I_0$ is problematic for the formula structure), leading to standard integrals like $\int dx$, $\int e^{ax}dx$, $\int \sin(x)dx$, $\int \cos(x)dx$, $\int \tan(x)dx$, or $\int \sec(x)dx$.

When to Use Reduction Formulae

Reduction formulae offer a systematic and efficient approach to integration in several scenarios:

• Integrands with Powers: They are ideal when the integrand contains a function raised to an integer power $n$ (e.g., $\sin^n x$, $x^n e^x$). Direct integration for larger $n$ can become extremely cumbersome.
• Repetitive Integration by Parts: If you find that applying integration by parts multiple times leads to a recurring pattern of integrals with decreasing powers, a reduction formula essentially encapsulates this pattern. This saves time and reduces the chance of algebraic errors.
• Systematic Evaluation: When you need to evaluate a family of integrals for various integer values of $n$, a reduction formula provides a clear, step-by-step procedure.
• Reaching a Known Integral: The process is designed to eventually reduce the original integral to one or more "base case" integrals that are well-known and can be solved directly (e.g., $\int e^x dx = e^x + C$, $\int \sin x \, dx = -\cos x + C$).

By pre-deriving the recursive relationship, reduction formulae streamline complex integration tasks, making them more manageable and less prone to error than repeated manual applications of techniques like integration by parts.