$\int_0^{\pi / 2} \frac{d x}{1+\tan ^3 x}$ is equal to :

$\int_0^\pi x \sin x \cos^4 x \, dx = $

If $I = \int_0^{\frac{\pi}{2}} \cos(\sin x) \,dx$,$J = \int_0^{\frac{\pi}{2}} \sin(\cos x) \,dx$,and $K = \int_0^{\frac{\pi}{2}} \cos x \,dx$,then:

If $\int_0^{\frac{\pi}{4}} \frac{\sin^2 x}{1+\sin x \cos x} dx = \frac{1}{a} \log_e\left(\frac{a}{3}\right) + \frac{\pi}{b \sqrt{3}}$,where $a, b \in N$,then $a+b$ is equal to ....................

$\int_{-\pi / 4}^{\pi / 4} x^3 \sin ^4(x) d x=$

If $F(x) = f(x) + f\left(\frac{1}{x}\right)$,where $f(x) = \int_{1}^{x} \frac{\log_{e} t}{1+t} dt$,then $F(e) = $