Evaluate the Integral integral of sin(x)^2 with respect to x

Problem

the integral of sine squared x d x

Solution

1. Identify the trigonometric identity for sine squared x to reduce the power, as it is not a basic integral.

2. Apply the identity sine squared x equals the fraction with numerator 1 minus cosine 2 x and denominator 2 to rewrite the integrand.

the integral of sine squared x d x equals the integral of the fraction with numerator 1 minus cosine 2 x and denominator 2 d x

3. Factor out the constant one half from the integral.

one half the integral of open paren 1 minus cosine 2 x close paren d x

4. Integrate the terms individually using the rule the integral of cosine a x d x equals sine a x over a

one half times open paren x minus sine 2 x over 2 close paren plus C

5. Distribute the constant to simplify the final expression.

x over 2 minus sine 2 x over 4 plus C

Final Answer

the integral of sine squared x d x equals x over 2 minus sine 2 x over 4 plus C

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