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
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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