Class 12 Integrals · Long Answer3 Marksindefinite_integration
Class 12 Maths · Integrals · Long Answer #155
Evaluate \(\int \frac{3 x + 5}{\sqrt{x^{2} - 8 x + 7}}\,dx\). Work on either interval x<1 or x>7.
✅ Step-by-Step Solution
Final Answer: \(3 \sqrt{x^{2} - 8 x + 7} + 17 \log{\left(\left|{x + \sqrt{x^{2} - 8 x + 7} - 4}\right| \right)}\) + C
- Let D=x²−8x+7=(x−1)(x−7), positive for x<1 or x>7. Write 3x+5=(3/2)(2x−8)+17.
- The integral is (3/2)∫D′/√D dx+17∫dx/√D=3√D+17∫dx/√[(x−4)²−9].
- Apply ∫du/√(u²−9)=ln|u+√(u²−9)| with u=x−4.
- Therefore the required integral is \(3 \sqrt{x^{2} - 8 x + 7} + 17 \log{\left(\left|{x + \sqrt{x^{2} - 8 x + 7} - 4}\right| \right)}\) + C.
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let D=x²−8x+7=(x−1)(x−7), positive for x<1 or x>7. Write 3x+5=(3/2)(2x−8)+17.
- [1 mark] The integral is (3/2)∫D′/√D dx+17∫dx/√D=3√D+17∫dx/√[(x−4)²−9].
- [1 mark] Apply ∫du/√(u²−9)=ln|u+√(u²−9)| with u=x−4.
- [0.5 marks] Therefore the required integral is \(3 \sqrt{x^{2} - 8 x + 7} + 17 \log{\left(\left|{x + \sqrt{x^{2} - 8 x + 7} - 4}\right| \right)}\) + C.
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