Question

integral of x^2/sqrt(x+5)​

Answer 1

Step-by-step explanation:

We have,

$\int \frac{ {x}^{2} }{ \sqrt{x + 5} } dx$

Let$\rm\:x+5=t^{2}$

$\rm\implies\:dx=2tdt$

So,

$\rm\int \frac{ \{ {t}^{2} - 5\} ^{2} }{ \sqrt{ {t}^{2} } } .2tdt$

$\rm = 2\int \frac{ \{ {t}^{2} - 5\} ^{2} }{ t} tdt$

$\rm = 2\int \{ {t}^{2} - 5\} ^{2} dt$

$\rm = 2\int ( {t}^{4} - 10 {t}^{2} + 25 ) dt$

$\rm = 2 \bigg \{ \frac{{t}^{5}}{5} - 10 \frac{ {t}^{3} }{3} + 25t + C \bigg \}$

$\rm = \frac{2}{5} {t}^{5}- \frac{ 20 }{3} {t}^{3}+50t + 2C$

$\rm = \frac{2}{5} { (\sqrt{x + 5} )}^{5}- \frac{ 20 }{3} {( \sqrt{x + 5}) }^{3}+50( \sqrt{x + 5} ) + k$

where, k=2C