PYQs Based Test - 04 : Definite and Indefinite Integrals

The integral $\underset{0}{\overset{1}{\int }}\frac{\mathrm{dx}}{\sqrt{1-\mathrm{x}}}$ is equal to _______

Given, $i=\sqrt{-1}$, the value of the definite integral, $I=\underset{0}{\overset{\pi /2}{\int }}\frac{\cos x+i\sin x}{\cos x-i\sin x}dx$ is:

Let f be a real-valued function of a real variable defined as f(x) = x – [x], where [x] denotes the largest integer less than or equal to x. The value of $\underset{0.25}{\overset{1.25}{\int }}f\left(x\right)dx$ is _______ (up to 2 decimal places).

The value (round off to one decimal place) of ${\int }_{-1}^{1}x{e}^{|x|}dx$ ______

If $f\left(x\right)=R\sin \left(\frac{\pi x}{2}\right)+S,f^{\prime }\left(\frac{1}{2}\right)=\sqrt{2}and{\int }_0^{1}f\left(x\right)dx=\frac{2R}{\pi }$, then the constants R and S are, respectively.