Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE

Evaluate the following definite integrals by definition:

${\int }_0^{1}xdx$

Evaluate the following definite integrals by definition:

${\int }_0^1{e}^{-x}dx$

Using definite integral find the sum of the following series:

If $\underset{n\to \infty }{\lim }\left[\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+\dots +\frac{1}{n+n}\right]=\log k$, find the value of $k$.

Using definite integral find the sum of the following series:

Using definite integral find the sum of the following series:

$\underset{n\to \infty }{\lim }\frac{1}{n}\left[\tan \frac{\pi }{4n}+\tan \frac{2\pi }{4n}+\dots +\tan \frac{n\pi }{4n}\right]$

If $f\left(x\right)={\int }_1^x\frac{t^{2}dt}{1+t^2},x>0,$ then find the value of $f^{\text{'}}\left(4\right)$ (Write the answer in the fraction form only)

If $f\left(x\right)={\int }_{\frac{1}{x^2}}^{x^2}\cos \sqrt{t}dt$ then the value of $f^{\text{'}}\left(1\right)=k\cos 1$, find $k$.

If for $x>0,{\int }_0^{x}g\left(t\right)\left[t^2+2t\right]dt=x^{2}g\left(x\right),$ then determine function $g\left(x\right)$.