Class 12 Integrals · Very Long Answer5 Marksdefinite_integration
Class 12 Maths · Integrals · Very Long Answer #115
Evaluate \(\int_0^{\pi/2}x\sin x\cos x/(\sin^4x+\cos^4x)dx\).
✅ Step-by-Step Solution
Final Answer: π²/16
- Let g(x)=sin x cos x/(sin⁴x+cos⁴x). Reflection x↦π/2−x leaves g unchanged.
- Adding the two forms of I gives 2I=(π/2)∫₀^(π/2)g(x)dx.
- With u=tan x, g(x)dx=u du/(1+u⁴), taking u from 0 to ∞.
- Set v=u², dv=2u du. Then ∫g dx=(1/2)∫₀^∞dv/(1+v²)=π/4. Hence 2I=π²/8.
- Therefore the integral equals π²/16.
📋 CBSE Stepwise Marking Scheme
- [1 mark] Let g(x)=sin x cos x/(sin⁴x+cos⁴x). Reflection x↦π/2−x leaves g unchanged.
- [1 mark] Adding the two forms of I gives 2I=(π/2)∫₀^(π/2)g(x)dx.
- [1 mark] With u=tan x, g(x)dx=u du/(1+u⁴), taking u from 0 to ∞.
- [1 mark] Set v=u², dv=2u du. Then ∫g dx=(1/2)∫₀^∞dv/(1+v²)=π/4. Hence 2I=π²/8.
- [1 mark] Therefore the integral equals π²/16.
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