Amit M Agarwal Solutions for Chapter: Definite Integrals, Exercise 4: Target Exercises

$\underset{n\to \infty }{\lim }{\left[\left(1+\frac{1^2}{n^2}\right)\left(1+\frac{2^2}{n^2}\right)\dots \left(1+\frac{n^2}{n^2}\right)\right]}^{\frac{1}{n}}$ is equal to

$\underset{n\to \infty }{\lim }\frac{1}{n^{p+1}}\left[1^p+2^p+\dots +n^p\right], p>-1,$ is equal to

$\underset{n\to \infty }{\lim }\left[\frac{1^2}{1^3+n^3}+\frac{2^2}{2^3+n^3}+\dots +\frac{1}{2n}\right]$ is equal to

The value of ${\int }_0^{\pi }{\cos }^{n}xdx,$ where $n$ is an even integer, is

If $I_n={\int }_0^{\frac{\pi }{2}}{x}^n\sin xdx,$ where $n>1,$ then

${\int }_0^a\frac{x^n}{\sqrt{ax-x^2}}dx$ is equal to

${\int }_0^{\pi }x{\sin }^{4}xdx$ is equal to

If ${\int }_0^{\frac{\pi }{2}}{\sin }^{6}xdx=\frac{5\pi }{32},$ then the value of ${\int }_{-\pi }^{\pi }\left(\sin x^6+\cos x^6\right)dx$ is