NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.6 Question 18

✅ Step-by-Step Solution

1. Work on each open interval where the original integrand is real and defined.
2. Recognize eˣ[g(x)+g′(x)] with g=\(\tan{\left(\frac{x}{2} \right)}\) and g′=\(\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{2} + \frac{1}{2}\).
3. The product rule makes its primitive eˣg(x); substitution into the original integrand verifies the match.
4. An antiderivative is F(x)=\(e^{x} \tan{\left(\frac{x}{2} \right)}\).
5. Differentiate this expression to check that F′(x)=\(\frac{\left(\sin{\left(x \right)} + 1\right) e^{x}}{\cos{\left(x \right)} + 1}\). Add an arbitrary constant C on each domain component.