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Class 12 Integrals · Very Long Answer5 Marksindefinite_integration

Class 12 Maths · Integrals · Very Long Answer #113

Evaluate \(\int \frac{x^{2}}{\left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(\frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}\) + C
  1. Write N=sin x−x cos x and D=x sin x+cos x.
  2. Differentiate: N′=x sin x and D′=x cos x.
  3. The quotient numerator is N′D−ND′=x sin x(x sin x+cos x)−x cos x(sin x−x cos x).
  4. The mixed terms cancel, leaving x²(sin²x+cos²x)=x². Therefore (N/D)′=x²/D², the integrand.
  5. Therefore the required integral is \(\frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] Write N=sin x−x cos x and D=x sin x+cos x.
  • [1 mark] Differentiate: N′=x sin x and D′=x cos x.
  • [1 mark] The quotient numerator is N′D−ND′=x sin x(x sin x+cos x)−x cos x(sin x−x cos x).
  • [1 mark] The mixed terms cancel, leaving x²(sin²x+cos²x)=x². Therefore (N/D)′=x²/D², the integrand.
  • [1 mark] Therefore the required integral is \(\frac{- x \cos{\left(x \right)} + \sin{\left(x \right)}}{x \sin{\left(x \right)} + \cos{\left(x \right)}}\) + C.

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