Definite Integration - Calculus - Mathematics Previous Year Questions

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

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

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

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