Class 12 Integrals · Long Answer3 Marksdefinite_integration
Class 12 Maths · Integrals · Long Answer #92
Evaluate \(\int_0^1x\ln(1+2x)dx\).
✅ Step-by-Step Solution
Final Answer: (3/8)ln3
- By parts, with u=ln(1+2x),dv=x dx, I=[x²ln(1+2x)/2]₀¹−∫₀¹x²/(1+2x)dx.
- Divide x²/(1+2x)=x/2−1/4+1/[4(1+2x)].
- The remainder integral is [x²/4−x/4+ln(1+2x)/8]₀¹=(ln3)/8; subtract from (ln3)/2.
- Therefore the integral equals (3/8)ln3.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] By parts, with u=ln(1+2x),dv=x dx, I=[x²ln(1+2x)/2]₀¹−∫₀¹x²/(1+2x)dx.
- [1 mark] Divide x²/(1+2x)=x/2−1/4+1/[4(1+2x)].
- [1 mark] The remainder integral is [x²/4−x/4+ln(1+2x)/8]₀¹=(ln3)/8; subtract from (ln3)/2.
- [0.5 marks] Therefore the integral equals (3/8)ln3.
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