∫(e^x(1+x) dx)/(cos^2(e^x x)) is equal to

The correct answer is Option (4) → \(\tan(xe^x)+c\), where \(c\) is constant of integration.

The numerator is the derivative of \(xe^x\), by the product rule:

    \( \displaystyle \frac{d}{dx}(xe^x) = e^x + xe^x = e^x(1 + x) \)

So, by substitution, let \(t = xe^x\), so \(dt = e^x(1+x)\,dx\):

    \( \displaystyle \int\frac{e^x(1+x)\,dx}{\cos^2(xe^x)} = \int\frac{dt}{\cos^2 t} \)

    \( \displaystyle = \int \sec^2 t\,dt = \tan t + c \)

    \(= \tan(xe^x) + c\)

NCERT (2026-27 edition):
Integration Using Trigonometric Identities