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Evaluate the following integral:

Question:

Evaluate the following integral:

$\int \frac{1}{(x-1) \sqrt{2 x+3}} d x$

Solution:

assume $2 x+3=t^{2}$

$\mathrm{d} \mathrm{x}=\mathrm{tdt}$

$\int \frac{d t}{\frac{t^{2}-3}{2}-1}$

$\int \frac{2 d t}{\left(t^{2}-5\right)}$

Using identity $\int \frac{\mathrm{dz}}{(\mathrm{z})^{2}-1}=\frac{1}{2} \log \left|\frac{z-1}{z+1}\right|+c$

$\frac{1}{\sqrt{5}} \log \left|\frac{t-\sqrt{5}}{t+\sqrt{5}}\right|+c$

$\frac{1}{\sqrt{5}} \log \left|\frac{\sqrt{(2 x+3)}-\sqrt{5}}{\sqrt{2 x+3}+\sqrt{5}}\right|+c$

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