CUET · MATHS · PYQ PAPER 2025
\(\int \frac{e^x(1+x) d x}{\cos ^2\left(e^x x\right)}\) is equal to:
- A \(\tan \left(e^x\right)+c\), where \(c\) is constant of integration.
- B \(-\cot \left(e^x x\right)+c\), where \(c\) is constant of integration.
- C \(\cot \left(e^x\right)+c\), where \(c\) is constant of integration.
- D \(\tan \left(x e^x\right)+c\), where \(c\) is constant of integration .
Answer & Solution
Correct Answer
(D) \(\tan \left(x e^x\right)+c\), where \(c\) is constant of integration .
Step-by-step Solution
Detailed explanation
Let \(u = x e^x\). Then \(du = (e^x + x e^x) dx = e^x(1+x) dx\). \(\int \frac{du}{\cos^2(u)} = \int \sec^2(u) du\) \(= \tan(u) + c\) \(= \tan(x e^x) + c\)
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