K. C. Sinha Solutions for Chapter: Definite Integrals and Their Properties, Exercise 3: EXERCISE

If $A={\int }_0^1{x}^{50}{\left(2-x\right)}^{50}dx; B={\int }_0^1{x}^{50}{\left(1-x\right)}^{50} dx$ and $\frac{A}{B}=2^n,$ then the value of $n$ is .....

If $f\left(4-x\right)=f\left(4+x\right)$ and $f\left(8-x\right)=f\left(8+x\right)$ and $f\left(x\right)$ is a function for which ${\int }_0^{8}f\left(x\right) dx=5$, then ${\int }_0^{200}f\left(x\right) dx$ is equal to

Let $f\left(x\right)={\int }_1^x\frac{dt}{t^3 {\left(1+t^3\right)}^{{\displaystyle \frac{1}{3}}}}$ and $\underset{x\to \infty }{\lim }f\left(x\right)=\frac{a^2-1}{a^3},$ then $a^{30}=$

Let $a={\int }_0^{\log 2}\frac{2e^{3x}+e^{2x}-1}{e^{3x}+e^{2x}-e^x+1} dx,$ then $4e^a=$

If $\alpha =\frac{\pi }{4020}$ and $\beta =\frac{2009\pi }{4020}$ and ${\int }_{\alpha }^{\beta }\frac{dx}{1+\tan x}=\frac{k\pi }{8040},$ then $k$ is equal to

The value of $I={\int }_0^3\left(\left[x\right]+\left[x+\frac{1}{3}\right]+\left[x+\frac{2}{3}\right]\right) dx,$ where $\left[·\right]$ denotes the greatest integer function, is equal to 

If $I={\int }_1^{10\pi }\left(\left[{\sec }^{-1}x\right]+\left[{\cot }^{-1}x\right]\right)dx,$ where $\left[x\right]$ denotes the integral part of $x,$ then the value of $n$ for which $I=n\pi -\sec 1$ is equal to

If $f\left(x\right)$ is a continuous function for all real values of $x$ and satisfies ${\int }_n^{n+1}f\left(x\right)dx=\frac{n^2}{2}, \forall n\in I,$ then $2{\int }_{-3}^{5}f\left(\left|x\right|\right) dx$ is equal to