Question:
The integral $\int \frac{e^x (1 + x)}{\cos^2(x e^x)} \, dx =$
The integral $\int \frac{e^x (1 + x)}{\cos^2(x e^x)} \, dx =$
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The derivative of $x e^x$ is $e^x(x+1)$. Seeing this combination in any integrand strongly suggests substituting $u = x e^x$.
Updated On: May 31, 2026
- $\tan(x e^x) + C$
- $-\tan(x e^x) + C$
- $\cot(x e^x) + C$
- $-\cot(x e^x) + C$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
We use the method of substitution by identifying the derivative of the angle argument $x e^x$ in the numerator.
Step 2: Meaning
The derivative of $x e^x$ with respect to $x$ using the product rule is $e^x(1 + x)$. This matches the numerator exactly.
Step 3: Analysis
Let: \[ u = x e^x \implies du = (1 \cdot e^x + x \cdot e^x) \, dx = e^x (1 + x) \, dx \] Substitute these into the integral: \[ I = \int \frac{du}{\cos^2 u} = \int \sec^2 u \, du \] Using the standard integration formula $\int \sec^2 u \, du = \tan u + C$: \[ I = \tan u + C = \tan(x e^x) + C \]
Step 4: Conclusion
The value of the indefinite integral is $\tan(x e^x) + C$. Final Answer: (A)
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