NCERT Class 12 · Ch 7 Exercise 7.3 · Q24
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.3 Question 24
Evaluate \(\int \frac{\left(x + 1\right) e^{x}}{\cos^{2}{\left(x e^{x} \right)}}\, dx\).
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: B: \(\tan{\left(x e^{x} \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- u=xeˣ, du=eˣ(1+x)dx; integrate sec²u.
- An antiderivative is F(x)=\(\tan{\left(x e^{x} \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{\left(x + 1\right) e^{x}}{\cos^{2}{\left(x e^{x} \right)}}\). Add an arbitrary constant C on each domain component.
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