Enter a trigonometric function (sin or cos) and a positive integer exponent n (typically n ≥ 2) into the calculator to display the standard reduction formula for the integral. This helps you reduce an integral with power n to one with power n − 2.

Reduction Formula Calculator

Enter a trigonometric function (sin or cos) and an exponent (n) to view the standard reduction formula.

Reduction Formula (Trig Integrals)

The following reduction formulas are commonly used to rewrite an integral of sinn(x) or cosn(x) in terms of an integral with the power reduced by 2. (A constant of integration + C is understood for indefinite integrals.)

\begin{aligned}
\int \sin^n(x)\,dx &= -\frac{\sin^{n-1}(x)\cos(x)}{n} + \frac{n-1}{n}\int \sin^{n-2}(x)\,dx \\
\int \cos^n(x)\,dx &= \frac{\cos^{n-1}(x)\sin(x)}{n} + \frac{n-1}{n}\int \cos^{n-2}(x)\,dx
\end{aligned}

Variables:

  • n is a positive integer exponent (typically n ≥ 2 for the reduction step).
  • x is the variable of integration.

To evaluate a specific integral, you typically apply the reduction formula repeatedly until the exponent becomes 0 or 1, then finish with the remaining basic integral.

What is a Reduction Formula?

A reduction formula is a recurrence relationship that expresses an integral (or other expression) with a parameter n in terms of the same type of integral with a smaller parameter (such as n − 2). For trigonometric integrals like ∫sinn(x)dx and ∫cosn(x)dx, reduction formulas are a standard method for systematically lowering the exponent.

How to Use the Reduction Formula Calculator

The following steps outline how to use the Reduction Formula Calculator.

  1. Choose whether you are working with sin or cos.
  2. Enter an integer exponent n (typically n ≥ 2).
  3. Click Calculate to display the standard reduction formula for ∫sinn(x)dx or ∫cosn(x)dx.
  4. If you are evaluating an integral, apply the reduction formula repeatedly to lower the exponent by 2 until you reach a base case (such as n = 0 or n = 1), then complete the remaining integral.
  5. Remember to include the constant of integration + C for indefinite integrals.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Function = sin

Exponent (n) = 5

Result (from the reduction formula): ∫sin5(x)dx = −(sin4(x)cos(x))/5 + (4/5)∫sin3(x)dx + C