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int(0)^( pi/4)log sin2 theta d theta...

int_(0)^( pi/4)log sin2 theta d theta

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Evaluate the following : int_(0)^(pi//2)sin theta d theta

STATEMENT 1: int_(0)^((pi)/(4))log(1+tan theta)d theta=(pi)/(8)log2 STATEMENT 2:int_(0)^((pi)/(2))log sin theta d theta=-pi log2

If int_(0)^((pi)/(2))log sin theta d(theta)=k, then find the value of int_(0)^((pi)/(2))((theta)/(sin theta))^(2)d(theta) in terms of k

STATEMENT 1: int_0^(pi/4)log(1+t a ntheta)d theta=pi/8log2. STATEMENT 2: int_0^(pi/2)logsin theta d theta =-pilog2.

int_(0)^( pi/2)sin^(2)theta d theta=

int_(0)^( pi/2)sin^(2)theta d theta=

int _(0)^(pi//4) ( 4 sin 2 theta d theta )/(sin^(4) theta +cos^(4) theta )=

int _(0)^(pi//4) ( 4 sin 2 theta d theta )/(sin^(4) theta +cos^(4) theta )=

int_(0)^( pi/2)sin^(3)theta d theta=