$$\int { x*\sqrt { (2-x) } } dx$$Substitution:$$u=\sqrt { (2-x) }$$$$\Rightarrow x=2-{ u }^{ 2 }$$$$\Rightarrow dx/du=-2u$$$$\Rightarrow dx=-2udu$$Damit:$$\int { x*\sqrt { (2-x) } } dx$$$$=\int { (2-{ u }^{ 2 })*u*(-2u)du }$$$$=\int { 2{ u }^{ 4 }-4{ u }^{ 2 }du }$$$$=\frac { 2 }{ 5 } { u }^{ 5 }-\frac { 4 }{ 3 } { u }^{ 3 }+C$$$$={ u }^{ 3 }(\frac { 2 }{ 5 } { u }^{ 2 }-\frac { 4 }{ 3 } )+C$$Rücksubsitution:$$={ (\sqrt { (2-x) } ) }^{ 3 }(\frac { 2 }{ 5 } (2-x)-\frac { 4 }{ 3 } )+C$$Ein bisschen Bruchrechnung:$$={ (2-x) }^{ \frac { 3 }{ 2 } }(-\frac { 2 }{ 5 } x-\frac { 8 }{ 15 } )+C$$
