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Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #72

Evaluate \(\int \frac{\sin^{6}{\left(x \right)} + \cos^{6}{\left(x \right)}}{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(- 3 x + \tan{\left(x \right)} - \cot{\left(x \right)}\) + C
  1. Use u³+v³=(u+v)³−3uv(u+v), with u=sin²x,v=cos²x. The numerator is 1−3sin²x cos²x.
  2. Dividing gives 1/(sin²x cos²x)−3=sec²x+csc²x−3.
  3. Integrate sec²x,csc²x and the constant to tan x,−cot x and −3x.
  4. Therefore the required integral is \(- 3 x + \tan{\left(x \right)} - \cot{\left(x \right)}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Use u³+v³=(u+v)³−3uv(u+v), with u=sin²x,v=cos²x. The numerator is 1−3sin²x cos²x.
  • [1 mark] Dividing gives 1/(sin²x cos²x)−3=sec²x+csc²x−3.
  • [1 mark] Integrate sec²x,csc²x and the constant to tan x,−cot x and −3x.
  • [0.5 marks] Therefore the required integral is \(- 3 x + \tan{\left(x \right)} - \cot{\left(x \right)}\) + C.

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