# {\int_{{0}}^{{a}}}{3}{x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}{\left.{d}{x}\right.}

$${\int_{{0}}^{{a}}}{3}{x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}{\left.{d}{x}\right.}$$

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Take the integral:
$$\int{3}{x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}{\left.{d}{x}\right.}$$
Factor out constants:
$$\int{3}{x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}{\left.{d}{x}\right.}$$
For the integrand $${x}^{2}\sqrt{{{a}^{2}-{x}^{2}}}$$, (assuming all variables are positive) substitute $${x}={a} \sin{{\left({u}\right)}}{\quad\text{and}\quad}{\left.{d}{x}\right.}={a} \cos{{\left({u}\right)}}{d}{u}$$. Then $$\sqrt{{{a}^{2}-{x}^{2}}}=\sqrt{{{a}^{2}-{a}^{2}{{\sin}^{2}{\left({u}\right)}}}}={a} \cos{{\left({u}\right)}}{\quad\text{and}\quad}{u}={{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}$$:
$$={3}{a}\int{a}^{3}{{\sin}^{2}{\left({u}\right)}}{{\cos}^{2}{\left({u}\right)}}{d}{u}$$
Factor out constants:
$$={3}{a}^{4}\int{{\sin}^{2}{\left({u}\right)}}{{\cos}^{2}{\left({u}\right)}}{d}{u}$$
Write $${{\cos}^{2}{\left({u}\right)}}{a}{s}{1}-{{\sin}^{2}{\left({u}\right)}}$$:
$$={3}{a}^{4}\int{{\sin}^{2}{\left({u}\right)}}{\left({1}-{{\sin}^{2}{\left({u}\right)}}\right)}{d}{u}$$
Expanding the integrand $${{\sin}^{2}{\left({u}\right)}}{\left({1}-{{\sin}^{2}{\left({u}\right)}}\right)}$$ gives $${{\sin}^{2}{\left({u}\right)}}-{{\sin}^{4}{\left({u}\right)}}$$:
$$={3}{a}^{4}\int{\left({{\sin}^{2}{\left({u}\right)}}-{{\sin}^{4}{\left({u}\right)}}\right)}{d}{u}$$
Integrate the sum term by term and factor out constants:
$$=-{3}{a}^{4}\int{{\sin}^{4}{\left({u}\right)}}{d}{u}+{3}{a}^{4}\int{{\sin}^{2}{\left({u}\right)}}{d}{u}$$
Use the reduction formula, $$\int{{\sin}^{m}{\left({u}\right)}}{d}{u}=-\frac{{ \cos{{\left({u}\right)}}{{\sin}^{{{m}-{1}}}{\left({u}\right)}}}}{{m}}+\frac{{{m}-{1}}}{{m}}\int{{\sin}^{{-{2}+{m}}}{\left({u}\right)}}{d}{u}$$, where m = 4:
$$=\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}+\frac{{{3}{a}^{4}}}{{4}}\int{{\sin}^{2}{\left({u}\right)}}{d}{u}$$
Write $${{\sin}^{2}{\left({u}\right)}}{a}{s}\frac{1}{{2}}-\frac{1}{{2}} \cos{{\left({2}{u}\right)}}$$:
$$=\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}+\frac{{{3}{a}^{4}}}{{4}}\int{\left(\frac{1}{{2}}-\frac{1}{{2}} \cos{{\left({2}{u}\right)}}\right)}{d}{u}$$
Integrate the sum term by term and factor out constants:
$$=\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}-\frac{{{3}{a}^{4}}}{{8}}\int \cos{{\left({2}{u}\right)}}{d}{u}+\frac{{{3}{a}^{4}}}{{8}}\int{1}{d}{u}$$
For the integrand cos(2 u), substitute $$s = 2 u$$ and $$ds = 2 du$$:
$$=\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}-\frac{{{3}{a}^{4}}}{{16}}\int \cos{{\left({s}\right)}}{d}{s}+\frac{{{3}{a}^{4}}}{{8}}\int{1}{d}{u}$$
The integral of $$\cos(s)$$ is $$\sin(s)$$:
$$=-\frac{3}{{16}}{a}^{4} \sin{{\left({s}\right)}}+\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}+\frac{{{3}{a}^{4}}}{{8}}\int{1}{d}{u}$$
The integral of 1 is u:
$$=-\frac{3}{{16}}{a}^{4} \sin{{\left({s}\right)}}+\frac{{{3}{a}^{4}{u}}}{{8}}+\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}+\text{constant}$$
Substitute back for $$s = 2 u$$:
$$=\frac{{{3}{a}^{4}{u}}}{{8}}+\frac{3}{{4}}{a}^{4}{{\sin}^{3}{\left({u}\right)}} \cos{{\left({u}\right)}}-\frac{3}{{8}}{a}^{4} \sin{{\left({u}\right)}} \cos{{\left({u}\right)}}+\text{constant}$$
Substitute back for $${u}={{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}$$:
$$=\frac{3}{{8}}{a}^{4}{{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}+\frac{3}{{4}}{a}^{4}{ \sin{{\left({{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}\right)}}^{3} \cos{{\left({{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}\right)}}}-\frac{3}{{8}}{a}^{4} \sin{{\left({{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}\right)}} \cos{{\left({{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}\right)}}+\text{constant}$$
Simplify using $$\cos{{\left({{\sin}^{{-{1}}}{\left({z}\right)}}\right)}}=\sqrt{{{1}-{z}^{2}}}{\quad\text{and}\quad} \sin{{\left({{\sin}^{{-{1}}}{\left({z}\right)}}\right)}}={z}$$:
$$=\frac{3}{{8}}{a}^{4}{{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}-\frac{3}{{8}}{a}^{2}{x}\sqrt{{{a}^{2}-{x}^{2}}}+\frac{3}{{4}}{x}^{3}\sqrt{{{a}^{2}-{x}^{2}}}+\text{constant}$$
Factor the answer a different way:
$$=\frac{1}{{8}}{\left({3}{a}^{4}{{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}+{3}{x}\sqrt{{{a}^{2}-{x}^{2}}}{\left({2}{x}^{2}-{a}^{2}\right)}\right)}+\text{constant}$$
Which is equivalent for restricted x and a values to:
$$=\frac{3}{{8}}\sqrt{{{a}^{2}-{x}^{2}}}{\left(-{a}^{2}{x}+\frac{{{a}^{3}{{\sin}^{{-{1}}}{\left(\frac{x}{{a}}\right)}}}}{\sqrt{{{1}-\frac{{x}^{2}}{{a}^{2}}}}}+{2}{x}^{3}\right)}+\text{constant}$$