Evaluate the following integrals: 3+2x-x^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\sqrt{3+2\text{x}-\text{x}^2}\text{dx}$

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Answer

$\int\sqrt{3+2\text{x}-\text{x}^2}\text{dx}=\int\sqrt{4-(\text{x}-1)^2}\text{dx}$

Let X - 1 = t, so that dx = dt

Thus,$\int\sqrt{3+2\text{x}-\text{x}^2}\text{dx}=\int\sqrt{4-\text{t}^2}\text{dt}$

$=\frac{1}{2}\text{t}\sqrt{4-\text{t}^2}+\frac{4}{2}\sin^{-1}\Big(\frac{\text{t}}{2}\Big)+\text{C}$

$=\frac{1}{2}(\text{x}-1)\sqrt{3+2\text{x}-\text{x}^2}+2\sin^{-1}\Big(\frac{\text{x}-1}{2}\Big)+\text{C}$

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