Evaluate the following integral:

Question:

Evaluate the following integral:

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

Solution:

assume $x+1=\frac{1}{t}$

$\mathrm{dx}=-\frac{1}{\mathrm{t}^{2}} \mathrm{dt}$

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

$-\int \frac{d t}{\sqrt{\frac{5}{4}-\left(t-\frac{1}{2}\right)^{2}}}$

Using identity $\int \frac{\mathrm{dx}}{\sqrt{\mathrm{a}^{2}-\mathrm{x}^{2}}}=\arcsin \left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{c}$

$-\arcsin \left(\frac{\left(t-\frac{1}{2}\right)}{\frac{\sqrt{5}}{2}}\right)+c$

Substituting $\mathrm{t}=\frac{1}{\mathrm{x}+1}$

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

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