Find: $\int e^x \left[ \frac{1}{(1 + x^2)^{3/2}} + \frac{x}{\sqrt{1 + x^2}} \right] dx$

$\frac{x e^x}{\sqrt{1 + x^2}} + C$

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 $f(x) = \frac{x}{\sqrt{1+x^2}}$

$f'(x) = \frac{\sqrt{1+x^2} \times (1) - x \times \frac{1}{2\sqrt{1+x^2}} \times 2x}{(\sqrt{1+x^2})^2}$

$f'(x) = \frac{\sqrt{1+x^2} - \frac{x^2}{\sqrt{1+x^2}}}{(1+x^2)}$

$f'(x) = \frac{1+x^2-x^2}{(1+x^2)^{3/2}} = \frac{1}{(1+x^2)^{3/2}}$

$∴I = \frac{xe^x}{\sqrt{1+x^2}} + C$