Question

${\int }_0^a{x}^2{(a-x)}^{3/2}dx$

Answer 1

Consider the given function,

$y={\int }_0^a{x}^2{(a-x)}^{\frac{3}{2}}dx$

$y={\int }_0^a{(x-a)}^2{(a-(a-x))}^{\frac{3}{2}}dx=$

$y={\int }_0^a{(x-a)}^2{\left(x\right)}^{\frac{3}{2}}dx$

$y={\int }_0^a(x^2+a^2-2ax){\left(x\right)}^{\frac{3}{2}}dx$

$y={\int }_0^a(x^2.x^{\frac{3}{2}}+a^2.x^{\frac{3}{2}}-2ax.x^{\frac{3}{2}})dx$

$y={\int }_0^a(x^{\frac{7}{2}}+a^2.x^{\frac{3}{2}}-2a{x}^{\frac{3}{2}})dx$

$y={\int }_0^a{x}^{\frac{7}{2}}+a{\int }_0^a{x}^{\frac{3}{2}}dx-2a{\int }_0^a{x}^{\frac{3}{2}}dx$

$={[\frac{2x^{\frac{9}{2}}}{9}+a^2.\frac{5x^{\frac{5}{2}}}{2}-2a.\frac{5x^{\frac{5}{2}}}{2}]}_0^a+C$

$=[\frac{2a^{\frac{9}{2}}}{9}+a^2.\frac{5a^{\frac{5}{2}}}{2}-2a.\frac{5a^{\frac{5}{2}}}{2}]+C$

$=\frac{41a^{\frac{9}{2}}}{2}+C$

Hence, this is the answer.