Question

$\displaystyle \red{\tt \int \limits^{x}_{0} { \theta}^{2} \sqrt{x - \theta} \: d \theta}$​

Answer 1

$\displaystyle \red{\tt \int \limits^{x}_{0} { \theta}^{2} \sqrt{x - \theta} \: d \theta}$

$\displaystyle \red{\tt \int \limits^{x}_{0} (x - { \theta}^{2}) \sqrt{x -(x - \theta)} \: d \theta}$

$\displaystyle \red{\tt \int \limits^{x}_{0} (x {}^{2} -2 { \theta}^{}x + { \theta}^{2} ) { \theta} ^{ \frac{1}{2} } \: d \theta}$

$\displaystyle \red{\tt \int \limits^{x}_{0} (x {}^{2} { \theta}^{ \frac{1}{2} } +2 { \theta}^{ \frac{3}{2} }x + { \theta} ^{ \frac{5}{2} } )\: d \theta}$

$\displaystyle \tt \red{ \left[ \frac{ {x}^{2} { \theta}^{ \frac{3}{2} } }{ \frac{3}{2} } - \frac{2x { \theta}^{ \frac{5}{2} } }{ \frac{5}{2} } + \frac{ { \theta}^{ \frac{7}{2} } }{ \frac{7}{2} } \right]^{x}_{0} }$

$\displaystyle \tt \red{ \frac{2}{3} {x}^{ \frac{7}{2} } - \frac{4}{5} {x}^{ \frac{7}{2} } + \frac{7}{2} {x}^{ \frac{7}{2} } }$

$\displaystyle \tt \red{ \frac{16}{105} {x}^{ \frac{7}{2} } }$