Mathematics

Integrate the following function: $$x\sqrt {x+2}$$


SOLUTION
$$\int { x\sqrt { x+2 } dx } $$
Let $$x+2=t$$
On differentiating w.r.t $$x$$ we have
$$\Rightarrow dx=dt$$
$$=\int { \left( t-2 \right) \sqrt { t }\; dt } $$
$$=\int { \left( t^{ 1+1/2 }-2t^{ 1/2 } \right) dt } $$
$$=\int { \left( t^{ 3/2 }-2t^{ 1/2 } \right) dt } $$
$$\Rightarrow \dfrac { t^{ 3/2+1 } }{ \dfrac { 3 }{ 2 } +1 } -\dfrac { 2{ t }^{ 1/2+1 } }{ \dfrac { 1 }{ 2 } +1 } =\dfrac { 2+\dfrac { 5 }{ 2 }  }{ 5 } -\dfrac { 2\times 2 }{ 3 } { t }^{ 3/2 }+c$$
                                       $$=\dfrac{2}{5}t^{5/2}-\dfrac{4}{3}t^{3/2}+c.$$
Hence, the answer is $$\dfrac{2}{5}t^{5/2}-\dfrac{4}{3}t^{3/2}+c.$$
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Subjective Medium Published on 17th 09, 2020
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