Class 12 Integrals · Very Long Answer5 Marksindefinite_integration

Class 12 Maths · Integrals · Very Long Answer #109

Evaluate \(\int \frac{x + 2}{\sqrt{x^{2} + 5 x + 6}}\,dx\). Work on either interval x<−3 or x>−2.

✅ Step-by-Step Solution

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

Let D=x²+5x+6=(x+2)(x+3). The real domain is x<−3 or x>−2. Write x+2=(2x+5)/2−1/2.
Split I=(1/2)∫D′/√D dx−(1/2)∫dx/√D.
The first term is √D. Complete the square D=(x+5/2)²−1/4.
The standard logarithmic primitive of 1/√[(x+5/2)²−1/4] is ln|x+5/2+√D|; combine the two terms.
Therefore the required integral is \(\sqrt{x^{2} + 5 x + 6} - \frac{\log{\left(\left|{x + \sqrt{x^{2} + 5 x + 6} + \frac{5}{2}}\right| \right)}}{2}\) + C.

[1 mark] Let D=x²+5x+6=(x+2)(x+3). The real domain is x<−3 or x>−2. Write x+2=(2x+5)/2−1/2.
[1 mark] Split I=(1/2)∫D′/√D dx−(1/2)∫dx/√D.
[1 mark] The first term is √D. Complete the square D=(x+5/2)²−1/4.
[1 mark] The standard logarithmic primitive of 1/√[(x+5/2)²−1/4] is ln|x+5/2+√D|; combine the two terms.
[1 mark] Therefore the required integral is \(\sqrt{x^{2} + 5 x + 6} - \frac{\log{\left(\left|{x + \sqrt{x^{2} + 5 x + 6} + \frac{5}{2}}\right| \right)}}{2}\) + C.

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