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Class 12 Integrals · Very Long Answer5 Marksindefinite_integration

Class 12 Maths · Integrals · Very Long Answer #173

Evaluate \(\int \frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{x \left(x + \sin{\left(x \right)}\right)}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\log{\left(\left|{x}\right| \right)} - \log{\left(\left|{x + \sin{\left(x \right)}}\right| \right)}\) + C
  1. Write sin x−x cos x=(x+sin x)−x(1+cos x).
  2. Dividing by x(x+sin x) gives the integrand 1/x−(1+cos x)/(x+sin x).
  3. The first term integrates to ln|x| on each interval avoiding zero.
  4. For the second let u=x+sin x, du=(1+cos x)dx, giving −ln|u|. Substitute back and combine.
  5. Therefore the required integral is \(\log{\left(\left|{x}\right| \right)} - \log{\left(\left|{x + \sin{\left(x \right)}}\right| \right)}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Write sin x−x cos x=(x+sin x)−x(1+cos x).
  • [1 mark] Dividing by x(x+sin x) gives the integrand 1/x−(1+cos x)/(x+sin x).
  • [1 mark] The first term integrates to ln|x| on each interval avoiding zero.
  • [1 mark] For the second let u=x+sin x, du=(1+cos x)dx, giving −ln|u|. Substitute back and combine.
  • [1 mark] Therefore the required integral is \(\log{\left(\left|{x}\right| \right)} - \log{\left(\left|{x + \sin{\left(x \right)}}\right| \right)}\) + C.

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