MHT-CET Full Test-9 Mathematics Que-22 Solution

Question-22

\int \frac{e^x\left(2-x^2\right)}{(1-x) \sqrt{1-x^2}} d x=

A)

e^x \sqrt{\frac{1+x}{1-x}}+c

B)

e^x \sqrt{\frac{1-x}{1+x}}+c

C)

e^x \sqrt{\frac{1-x}{1+x^2}}+c

D) None of these

Answer: A)

Explanation

I = \int \frac{e^x(2 - x^2)}{(1 - x) \sqrt{1 - x^2}} \, dx \Rightarrow I = \int \frac{e^x(1 + 1 - x^2)}{(1 - x) \sqrt{1 - x^2}} \, dx = \int e^x \left[\frac{1 - x^2}{(1 - x) \sqrt{1 - x^2}} + \frac{1}{(1 - x) \sqrt{1 - x^2}}\right] \, dx = \int e^x \left[\frac{\sqrt{1 - x^2}}{(1 - x)} + \frac{1}{(1 - x) \sqrt{1 - x^2}}\right] \, dx = \int e^x \left[ \underbrace{\sqrt{\frac{1 + x}{1 - x}}}_{f(x)} + \underbrace{\frac{1}{(1 - x) \sqrt{1 - x^2}}}_{f'(x)} \right] \, dx \Rightarrow I = e^x \sqrt{\frac{1 + x}{1 - x}} + c \left[\because \int e^x \left[f(x) + f'(x)\right] \, dx = e^x f(x) + c \right]