Evaluate the following integral:

Question:

Evaluate the following integral:

$\int \frac{x}{\left(x^{2}+4\right) \sqrt{x^{2}+9}} d x$

Solution:

assume $x^{2}+9=u^{2}$

$x d x=u d u$

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

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

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

$\frac{1}{2 \sqrt{5}} \log \left|\frac{u-\sqrt{5}}{u+\sqrt{5}}\right|+c$

Substituting $u=\sqrt{9+x^{2}}$

$\frac{1}{2 \sqrt{5}} \log \left|\frac{\sqrt{9+x^{2}}-\sqrt{5}}{\sqrt{9+x^{2}}+\sqrt{5}}\right|+c$

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