Evaluate ∫ (e^x(1+x))/(cos²(xe^x)) , dx — JEE Mathematics

Let $u = xe^x$.
Differentiate with respect to $x$ using the product rule:

$$du = (1 \cdot e^x + x \cdot e^x) \, dx = e^x(1 + x) \, dx$$

This matches the numerator perfectly. Substitute $u$ into the integral:

$$I = \int \frac{du}{\cos^2 u} = \int \sec^2 u \, du = \tan u + C$$

Substitute back $u = xe^x$:

$$I = \tan(xe^x) + C$$

Answer: $\tan(xe^x) + C$