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Class 12 Integrals · Very Long Answer5 Marksindefinite_integration

Class 12 Maths · Integrals · Very Long Answer #110

Evaluate \(\int \frac{1}{\sin^{4}{\left(x \right)} + \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{4}{\left(x \right)}}\,dx\). Work on an interval with cos x≠0; extend across such points by matching constants.

✅ Step-by-Step Solution

Final Answer: \(\frac{\sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \left(2 \tan{\left(x \right)} - 1\right)}{3} \right)} + \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 \tan{\left(x \right)} + 1\right)}{3} \right)}\right)}{3}\) + C
  1. The original denominator equals 1−sin²x cos²x≥3/4, so the integrand is continuous on R. On an interval with cos x≠0 set u=tan x, dx=du/(1+u²).
  2. Dividing the original numerator and denominator by cos⁴x gives I=∫(1+u²)/(u⁴+u²+1)du.
  3. Factor u⁴+u²+1=(u²+u+1)(u²−u+1). The fraction is (1/2)[1/(u²+u+1)+1/(u²−u+1)].
  4. Complete the squares (u±1/2)²+3/4 and integrate both quadratic reciprocals. Substitute u=tan x; adjacent constants can be matched to extend continuously across cosine zeros.
  5. Therefore the required integral is \(\frac{\sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \left(2 \tan{\left(x \right)} - 1\right)}{3} \right)} + \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 \tan{\left(x \right)} + 1\right)}{3} \right)}\right)}{3}\) + C.

📋 CBSE Stepwise Marking Scheme

  • [1 mark] The original denominator equals 1−sin²x cos²x≥3/4, so the integrand is continuous on R. On an interval with cos x≠0 set u=tan x, dx=du/(1+u²).
  • [1 mark] Dividing the original numerator and denominator by cos⁴x gives I=∫(1+u²)/(u⁴+u²+1)du.
  • [1 mark] Factor u⁴+u²+1=(u²+u+1)(u²−u+1). The fraction is (1/2)[1/(u²+u+1)+1/(u²−u+1)].
  • [1 mark] Complete the squares (u±1/2)²+3/4 and integrate both quadratic reciprocals. Substitute u=tan x; adjacent constants can be matched to extend continuously across cosine zeros.
  • [1 mark] Therefore the required integral is \(\frac{\sqrt{3} \left(\operatorname{atan}{\left(\frac{\sqrt{3} \left(2 \tan{\left(x \right)} - 1\right)}{3} \right)} + \operatorname{atan}{\left(\frac{\sqrt{3} \left(2 \tan{\left(x \right)} + 1\right)}{3} \right)}\right)}{3}\) + C.

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