Question

If $f\left(x\right)=sin x,g\left(x\right)=cos⁡x$ and $h\left(x\right)=cos\left(cos x\right),$ then the integral $I =\int f(g(x)) \cdot f(x) \cdot h(x) d x$ simplifies to $-\lambda \left(sin\right)^{2} \left(cos ⁡ x\right)+C$ (where, $C$ is the constant of integration). The value of $\lambda $ is equal to

Answer 1

Given integral, $Ι=\displaystyle \int s i n \left(cos x\right) . sin ⁡ x . c o s \left(cos ⁡ x\right) d x$
Let, $sin\left(cos x\right)=t\Rightarrow -cos\left(cos ⁡ x\right).sin ⁡ xdx=dt$
$\therefore Ι=\displaystyle \int - t d t=-\frac{t^{2}}{2}+C$
$=C-\frac{s i n^{2} \left(cos x\right)}{2}$