Evaluate the following integrals: 1 (x-1) 2x+3 dx - Vidyadip

Question

Evaluate the following integrals:
$\int \frac{1}{(\text{x}-1)\sqrt{2\text{x}+3}}\text{ dx}$

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Answer

Let $\text{I}=\int \frac{1}{(\text{x}-1)\sqrt{2\text{x}+3}}\text{ dx}$
Let $2\text{x}+3=\text{t}^2$
$2\text{dx}=2\text{tdt}$
$\therefore\ \text{I}=\int\frac{\text{t dt}}{\big(\frac{\text{t}^2-3}{2}-1\big)\text{t}}$
$=2\int\frac{\text{dt}}{\text{t}^2-5}$
$=\frac{2}{2\sqrt{5}}\log\Big|\frac{\text{t}-\sqrt{5}}{\text{t}+\sqrt{5}}\Big|+\text{C}$
Thus, $\text{I}=\frac{1}{\sqrt{5}}\log\bigg|\frac{\sqrt{2\text{x}+3}-\sqrt{5}}{\sqrt{2\text{x}+3}+\sqrt{5}}\bigg|+\text{C}$

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