Definite Integral of sin²(x) from 0 to π

Published on October 13, 2024

Calculus

Calculate the definite integral of sin²(x) from 0 to π. Find the solution to this trigonometric integral problem, demonstrating the use of the identity sin²(x) = (1-cos(2x))/2. The answer is π/2.

Question

Calculate the definite integral of g(x) = sin^2(x) from 0 to π

Answer

Let’s break down how to calculate the definite integral of g(x) = sin²(x) from 0 to π.

1. Recall the trigonometric identity:

2. Substitute the identity into the integral:

3. Apply linearity of the integral:

4. Evaluate the integrals:

The integral of 1 is simply x:
For the second integral, use substitution. Let u = 2x, then du = 2 dx:

5. Combine the results:

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