Evaluate: `int_0^(pi/4)(tan^(-1)((2cos^2theta)/(2-sin2theta)))sec^2theta d theta dot`

If I_(1)=int_(0)^(1)(dx)/(e^(x)(1+x)) and I_(2)=int_(0)^(pi//4)(e^(tan^(7)theta)sintheta)/((2-tan^(2)theta)cos^(3)theta d theta ,then find the value of (l_(1))/(l_(2)) .

Prove: (cos theta)/(1+sin theta) = sec theta - tan theta

(1+sin2theta+cos2theta)/(1+sin2theta-cos2theta)=?

Evaluate int_(0)^((pi)/(2))(cos theta d theta)/((sin theta +2)(sin theta +3)) .

(1-Sin^(2)theta)(sec^(2)theta)= ………… .

Evaluate: int_(0)^(pi//2) sqrtsinthetacos^(5)theta d theta

Let f beintegrable over [0,a] for any real value of a . If I_(1)=int_(0)^(pi//2)cos theta f(sin theta +cos^(2) theta) d theta and I_(2)=int_(0)^(pi//2) sin 2 theta f(sin theta+cos^(2) theta) d theta , then

(1)/(sin^(2)theta)-(cos^(2)theta)/(sin^(2) theta) =___.

Prove that: (cos2theta)/(1+sin2theta)=tan(pi/4-theta)