Definite Integration · Mathematics · VITEEE

Let $f$ be a non-negative function defined on the interval $[0,1]$. If $\int_0^x \sqrt{1-(f(t))^2} d t =\int_0^x f(t) d t, 0 \leq x \leq 1$ and $f(0)=0$, then

$$ \int_0^\pi\left[\cos ^2\left(\frac{3 \pi}{8}-\frac{x}{4}\right)-\cos ^2\left(\frac{11 \pi}{8}+\frac{x}{4}\right)\right] d x$$ equals to

The value of $$\int_\lambda^{\lambda+\pi / 2}\left(\cos ^4 x+\sin ^4 x\right) d x$$ is

$$\lim _\limits{n \rightarrow \infty}\left(\frac{(n+1)(n+2) \ldots 3 n}{n^{2 n}}\right)^{1 / n}$$ is equal to

If $$X \phi(x)=\int_\limits5^x 3 t^2-2 \phi(t) d t, x>-2$$ and $$\phi(0)=4$$, then $$\phi(2)$$ is

The value of $$\int\limits_0^{\pi / 2} \frac{d x}{1+\tan x}$$ is