NCERT Class 12 · Ch 7 Exercise 7.3 · Q20
NCERT Class 12 Maths · Chapter 7 Integrals · Exercise 7.3 Question 20
Evaluate \(\int \frac{\cos{\left(2 x \right)}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\, dx\).
💡 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: \(\log{\left(\left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right| \right)}\) + C.
- Work on each open interval where the original integrand is real and defined.
- Factor cos 2x=(cos x−sin x)(cos x+sin x); use the logarithmic derivative.
- An antiderivative is F(x)=\(\log{\left(\left|{\sin{\left(x \right)} + \cos{\left(x \right)}}\right| \right)}\).
- Differentiate this expression to check that F′(x)=\(\frac{\cos{\left(2 x \right)}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\). Add an arbitrary constant C on each domain component.
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