Class 12 Integrals · Very Long Answer5 Marksindefinite_integration

Class 12 Maths · Integrals · Very Long Answer #107

Evaluate \(\int \frac{5 x + 3}{\sqrt{x^{2} + 4 x + 10}}\,dx\). Work on each connected open interval where the real integrand is defined.

✅ Step-by-Step Solution

Final Answer: \(5 \sqrt{x^{2} + 4 x + 10} - 7 \log{\left(x + \sqrt{x^{2} + 4 x + 10} + 2 \right)}\) + C

Let D=x²+4x+10. Express the numerator as 5x+3=(5/2)(2x+4)−7.
Split I=(5/2)∫D′/√D dx−7∫dx/√D.
The first integral is 2√D by u=D. For the second, complete D=(x+2)²+6.
Using ∫du/√(u²+6)=ln|u+√(u²+6)| gives I=5√D−7ln|x+2+√D|; the log argument is positive.
Therefore the required integral is \(5 \sqrt{x^{2} + 4 x + 10} - 7 \log{\left(x + \sqrt{x^{2} + 4 x + 10} + 2 \right)}\) + C.

[1 mark] Let D=x²+4x+10. Express the numerator as 5x+3=(5/2)(2x+4)−7.
[1 mark] Split I=(5/2)∫D′/√D dx−7∫dx/√D.
[1 mark] The first integral is 2√D by u=D. For the second, complete D=(x+2)²+6.
[1 mark] Using ∫du/√(u²+6)=ln|u+√(u²+6)| gives I=5√D−7ln|x+2+√D|; the log argument is positive.
[1 mark] Therefore the required integral is \(5 \sqrt{x^{2} + 4 x + 10} - 7 \log{\left(x + \sqrt{x^{2} + 4 x + 10} + 2 \right)}\) + C.

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