NCERT Class 12 · Ch 7 Miscellaneous Exercise · Q20
NCERT Class 12 Maths · Chapter 7 Integrals · Miscellaneous Exercise Question 20
Evaluate \(\int \frac{\left(\sin{\left(2 x \right)} + 2\right) e^{x}}{\cos{\left(2 x \right)} + 1}\, dx\).
💡 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: \(e^{x} \tan{\left(x \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Use 1+cos 2x=2cos²x to rewrite the fraction as sec²x+tan x.
- This is the product derivative of eˣtan x.
- An antiderivative is F(x)=\(e^{x} \tan{\left(x \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{\left(\sin{\left(2 x \right)} + 2\right) e^{x}}{\cos{\left(2 x \right)} + 1}\). Add an arbitrary constant C on each domain component.
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