Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \frac{1}{\sqrt{3 x^{2}+5 x+7}} d x$

Solution:

let $I=\int \frac{1}{\sqrt{3 x^{2}+5 x+7}} d x$

$=\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{x^{2}+\frac{5}{3} x+\frac{7}{3}}} d x$

$=\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{x^{2}+2 x\left(\frac{5}{6}\right)+\left(\frac{5}{6}\right)^{2}-\left(\frac{5}{6}\right)^{2}+\frac{7}{3}}} d x$

$=\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{\left(x+\frac{5}{6}\right)^{2}-\frac{59}{36}}} d x$

$\operatorname{let}\left(x+\frac{5}{6}\right)=t$

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

$I=\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{t^{2}-\left(\frac{\sqrt{59}}{6}\right)^{2}}} d t$

$=\frac{1}{\sqrt{3}} \log \left|t+\sqrt{t^{2}-\left(\frac{\sqrt{59}}{6}\right)}\right|+c\left[\right.$ since $\int \frac{1}{\sqrt{x^{2}-a^{2}}} d x=\log \left|x+\sqrt{x^{2}-a^{2}}\right|+c$

$I=\frac{1}{\sqrt{3}} \log \left|x+\frac{5}{6}+\sqrt{\left(x+\frac{5}{6}\right)^{2}-\left(\frac{\sqrt{59}}{6}\right)^{2}}\right|+c$

$I=\frac{1}{\sqrt{3}} \log \left|x+\frac{5}{6}+\sqrt{x^{2}+\frac{5 x}{3}+\frac{7}{3}}\right|+c$

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