Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #63
Evaluate \(\int \frac{1}{\sqrt{x} \left(\sqrt{x} + 1\right) \left(\sqrt{x} + 2\right)}\,dx\). Work for x>0.
✅ Step-by-Step Solution
Final Answer: \(2 \log{\left(\frac{\sqrt{x} + 1}{\sqrt{x} + 2} \right)}\) + C
- Let u=√x, so dx/√x=2du.
- The integral is 2∫du/[(u+1)(u+2)]=2∫[1/(u+1)−1/(u+2)]du.
- Integrate to 2ln|(u+1)/(u+2)| and replace u by √x; the ratio is positive.
- Therefore the required integral is \(2 \log{\left(\frac{\sqrt{x} + 1}{\sqrt{x} + 2} \right)}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let u=√x, so dx/√x=2du.
- [1 mark] The integral is 2∫du/[(u+1)(u+2)]=2∫[1/(u+1)−1/(u+2)]du.
- [1 mark] Integrate to 2ln|(u+1)/(u+2)| and replace u by √x; the ratio is positive.
- [0.5 marks] Therefore the required integral is \(2 \log{\left(\frac{\sqrt{x} + 1}{\sqrt{x} + 2} \right)}\) + C.
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