Class 12 Integrals · Very Long Answer5 Marksindefinite_integration

Class 12 Maths · Integrals · Very Long Answer #120

Evaluate \(\int \frac{2 x + 1}{\sqrt{- x^{2} + 2 x + 3}}\,dx\). Work for −1<x<3.

✅ Step-by-Step Solution

Final Answer: \(- 2 \sqrt{- x^{2} + 2 x + 3} + 3 \operatorname{asin}{\left(\frac{x}{2} - \frac{1}{2} \right)}\) + C

Let D=3+2x−x²=4−(x−1)²; the real domain is −1<x<3.
Express 2x+1=−(2−2x)+3=−D′+3, splitting the integral into −∫D′/√D dx+3∫dx/√D.
The first term is −2√D by u=D.
In the second let v=(x−1)/2. Then ∫dx/√[4−(x−1)²]=arcsin v; combine the two terms.
Therefore the required integral is \(- 2 \sqrt{- x^{2} + 2 x + 3} + 3 \operatorname{asin}{\left(\frac{x}{2} - \frac{1}{2} \right)}\) + C.

[1 mark] Let D=3+2x−x²=4−(x−1)²; the real domain is −1<x<3.
[1 mark] Express 2x+1=−(2−2x)+3=−D′+3, splitting the integral into −∫D′/√D dx+3∫dx/√D.
[1 mark] The first term is −2√D by u=D.
[1 mark] In the second let v=(x−1)/2. Then ∫dx/√[4−(x−1)²]=arcsin v; combine the two terms.
[1 mark] Therefore the required integral is \(- 2 \sqrt{- x^{2} + 2 x + 3} + 3 \operatorname{asin}{\left(\frac{x}{2} - \frac{1}{2} \right)}\) + C.

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