Question

Integrate using u substitution method

$\displaystyle\int x \sqrt{x + 2} \: \: dx$

Need step by step explained answer.​

Answer 1

SolutioN :

$\rightarrow \tt \: \: \int x \sqrt{x + 2} \, dx$

Let,

$\bullet \tt \: \: \sqrt{x + 2} = u$

$\rightarrow \tt \: \: x + 2 = {u}^{2} \: \: \: \: \: \: \: \: \: \: \: \: - (1)$

$\rightarrow \tt \: \: x = {u}^{2} - 2 \: \: \: \: \: \: \: \: \: \: \: \: - (2)$

$\rightarrow \tt \: \: dx = 2u\,du \: \: \: \: \: \: \: \: \: \: \: \: - (3)$

⚝ Now, Substitute the value from ( 1 ) , ( 2 ) and ( 3 )

$\rightarrow \tt \: \: \int ( {u}^{2} - 2)(u)(2u)\, du$

$\rightarrow \tt \: \: \int (2 {u}^{4} - 4 {u}^{2} )\, du$

$\rightarrow \tt \: \: 2\int {u}^{4} \,du - 4\int {u}^{2} \, du$

$\rightarrow \tt \: \: \bigg( \frac{2}{5} \times {u}^{5}\bigg) - \bigg(\frac{4}{3} \times {u}^{3} \bigg) + c$

$\rightarrow \tt \: \: \bigg( \frac{2}{5} \times {(x + 2)}^{ \frac{5}{2} } \bigg)- \bigg( \frac{4}{3} \times {(x + 2)}^{ \frac{3}{2} } \bigg)+ c$