Evaluate the following integrals: 1 (x-1) x+2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int \frac{1}{(\text{x}-1)\sqrt{\text{x}+2}}\text{ dx}$

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Answer

Let $\text{I}=\int \frac{1}{(\text{x}-1)\sqrt{\text{x}+2}}\text{ dx}$
Let $\text{x}+2=\text{t}^2$
$\therefore\ \text{I}=\int\frac{2\text{tdt}}{(\text{t}^2-3)\text{t}}$
$=2\int\frac{\text{dt}}{\text{t}^2-3}$
$=\frac{2}{\sqrt{3}}\log\Big|\frac{\text{t}-\sqrt{3}}{\text{t}+\sqrt{3}}\Big|+\text{C}$
Thus, $\text{I}=\frac{1}{\sqrt{3}}\log\bigg|\frac{\sqrt{\text{x}-2}-\sqrt{3}}{\sqrt{\text{x}+2}+\sqrt{3}}\bigg|+\text{C}$

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