For $f(x) = x^4 + |x|$,let $I_1 = \int_{0}^{\pi} f(\cos x) dx$ and $I_2 = \int_{0}^{\frac{\pi}{2}} f(\sin x) dx$. Then $\frac{I_1}{I_2}$ is equal to

$\int_0^{\pi / 2} \frac{200 \sin x+100 \cos x}{\sin x+\cos x} d x$ is equal to (in $\pi$)

Let $f(x)$ be a continuous periodic function with period $T$. Let $I = \int_{a}^{a+T} f(x) \, dx$. Then

If $f$ is a continuous function and $f(x+T)=f(x)$ for all $x \in R$,it is given that $\int_0^{NT} f(t) dt = N \int_0^T f(t) dt$ (where $N$ is a natural number). Then,evaluate $\int_0^{50\pi} \sqrt{1-\cos 2x} dx$.

If $\int_{0}^{1} x^{m} (1 - x)^{n} dx = k \int_{0}^{1} x^{n} (1 - x)^{m} dx$,then the value of $k$ equals

$\int_0^{\pi / 2} \frac{\sin x}{1+\cos x+\sin x} d x=$