Question

refer to the attachment and answer correctly​

Answer 1

Answer:

$\Large{\boxed{\underline{\overline{\mathfrak{\star \: AnSwer :- \: \star}}}}}$

Given Expression,

$\displaystyle \sf \: l = \int \: {x}^{2} \sqrt{1 + x} \: dx$

Assuming x + 1 = t

Differentiating both sides of the equation,

$\sf \: \: dx = dt$

The expression would be re written as :

$\longrightarrow \displaystyle \sf \: l = \int \: (t - 1) {}^{2} \sqrt{t} \: dt \\ \\ \longrightarrow \displaystyle \sf \: l = \int( {t}^{2} - 2t + 1) \sqrt{t} .dt \\ \\ \longrightarrow \displaystyle \sf \: l = \int( {t}^{ \frac{5}{2} } - 2t {}^{ \frac{3}{2} } + \sqrt{t} ).dt$

We know that,

$\displaystyle \: \sf \: \int {x}^{n} = \dfrac{ {x}^{n + 1} }{n + 1} + c$

Thus,

$\longrightarrow \: \sf \: l = \dfrac{2 {t}^{ \frac{7}{2} } }{7} - \dfrac{4 {t}^{ \frac{5}{2} } }{5} + \dfrac{2t {}^{ \frac{3}{2} } }{3} + c \\ \\ \longrightarrow \: \boxed{ \boxed{ \sf l = \dfrac{2 \sqrt{(x + 1) {}^{7} } }{7} - \dfrac{4 \sqrt{(x + 1) {}^{5} } }{5} + \dfrac{2 \sqrt{(x + 1) {}^{3} } }{3} + c }}$