Find: $\int e^x \left[ \frac{1}{(1 + x^2)^{3/2}} + \frac{x}{\sqrt{1 + x^2}} \right] dx$
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $\frac{x e^x}{\sqrt{1 + x^2}} + C$
Let $I = \int e^x \left[ \frac{1}{(1+x^2)^{3/2}} + \frac{x}{\sqrt{1+x^2}} \right] dx$ ...(i)
We know that
$\int e^x [f(x) + f'(x)] dx = [e^x f(x) + C]$
Let $f(x) = \frac{x}{\sqrt{1+x^2}}$
$f'(x) = \frac{\sqrt{1+x^2} \times (1) - x \times \frac{1}{2\sqrt{1+x^2}} \times 2x}{(\sqrt{1+x^2})^2}$
$f'(x) = \frac{\sqrt{1+x^2} - \frac{x^2}{\sqrt{1+x^2}}}{(1+x^2)}$
$f'(x) = \frac{1+x^2-x^2}{(1+x^2)^{3/2}} = \frac{1}{(1+x^2)^{3/2}}$
$∴I = \frac{xe^x}{\sqrt{1+x^2}} + C$