# \int\frac{{x}^{2}}{{x}}\frac{{\left({4}+{9}{x}^{2}\right)}^{1}}{{2}}{\left.{d}{x}\right.}

$$\int\frac{{x}^{2}}{{x}}\frac{{\left({4}+{9}{x}^{2}\right)}^{1}}{{2}}{\left.{d}{x}\right.}$$

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Take the integral:
$$\int{x}\sqrt{{{9}{x}^{2}+{4}}}{\left.{d}{x}\right.}$$
For the integrand x $$\sqrt{{{9}{x}^{2}+{4}}}$$, substitute $${u}={9}{x}^{2}+{4}{\quad\text{and}\quad}{d}{u}={18}{x}{\left.{d}{x}\right.}$$:
$$=\frac{1}{{18}}\int\sqrt{{{u}}}{d}{u}$$
The integral of $$\sqrt{{{u}}}$$ is $$\frac{{{2}{u}^{{\frac{3}{{2}}}}}}{{3}}$$:
$$=\frac{{u}^{{\frac{3}{{2}}}}}{{27}}+\text{constant}$$
Substitute back for $${u}={9}{x}^{2}+{4}$$:
$$=\frac{1}{{27}}{\left({9}{x}^{2}+{4}\right)}^{{\frac{3}{{2}}}}+\text{constant}$$

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