Question

Integrate the following functions w.r.t $X:3^{{\cos }^{2}x}\sin 2x$

Answer 1

Let $I=\int 3^{{\cos }^{2}x}\sin 2xdx$

Put ${\cos }^{2}x=t$

$\therefore \left[2\cos x\frac{d}{dx}\right(\cos x\left)\right]dx=dt$

$\therefore -2\sin x=\cos xdx=dt$

$\therefore \sin 2xdx=-dt$

$I=-\int 3^{t}dt$

$=-\frac{1}{\log 3}{.3}^t+c$

$=-\dfrac{1}{\log 3}.3^{\cos^2 x}

+c$