Question

Evaluate the definite integral: $∫_0^π(sin^2\frac{x}{2}-cos^2\frac{x}{2})dx$

Answer 1

Let I=$∫_0^π(sin^2\frac{x}{2}-cos^2\frac{x}{2})dx$

$=∫_0^π(cos^2\frac{x}{2}-sin^2\frac{x}{2})dx$

$=-∫_0^πcosx dx$

$∫cosx dx=sin x=F(x)$

By second fundamental theorem of calculus,we obtain

$I=F(π)-F(0)$

$=sinπ-sin0$

$=0$