Class 12 Integrals · Long Answer3 Marksindefinite_integration

Class 12 Maths · Integrals · Long Answer #157

Evaluate \(\int \frac{x + 2}{\sqrt{\left(x - 3\right) \left(x - 2\right)}}\,dx\). Work on either interval x<2 or x>3.

✅ Step-by-Step Solution

Final Answer: \(\sqrt{x^{2} - 5 x + 6} + \frac{9 \log{\left(\left|{x + \sqrt{x^{2} - 5 x + 6} - \frac{5}{2}}\right| \right)}}{2}\) + C

Let D=(x−2)(x−3)=x²−5x+6. Write x+2=(2x−5)/2+9/2.
The first part (1/2)∫D′/√D dx is √D. For the second, D=(x−5/2)²−1/4.
Use the standard logarithmic primitive for 1/√[(x−5/2)²−1/4], with coefficient 9/2.
Therefore the required integral is \(\sqrt{x^{2} - 5 x + 6} + \frac{9 \log{\left(\left|{x + \sqrt{x^{2} - 5 x + 6} - \frac{5}{2}}\right| \right)}}{2}\) + C.

[0.5 marks] Let D=(x−2)(x−3)=x²−5x+6. Write x+2=(2x−5)/2+9/2.
[1 mark] The first part (1/2)∫D′/√D dx is √D. For the second, D=(x−5/2)²−1/4.
[1 mark] Use the standard logarithmic primitive for 1/√[(x−5/2)²−1/4], with coefficient 9/2.
[0.5 marks] Therefore the required integral is \(\sqrt{x^{2} - 5 x + 6} + \frac{9 \log{\left(\left|{x + \sqrt{x^{2} - 5 x + 6} - \frac{5}{2}}\right| \right)}}{2}\) + C.

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