Question

evaluate this integral and ill give medal

Answer 1

\[\int\limits_{}^{}\frac{ (2+\sqrt{x })^2dx }{ \sqrt{x} }\]

Answer 2

looks like function times its derivative
as
\[y \prime =\sqrt{x} = \frac{ 1 }{ 2*\sqrt{x} }\]

Answer 3

\[d(2 + \sqrt{x}) = \frac{ 1 }{ 2 \sqrt{x} } dx\]

Answer 4

Is it clear?

Answer 5

so \[= (\frac{ 1 }{ 3 })\frac{ (2+\sqrt{x})^3 }{ \sqrt{x} }(\frac{ 1 }{ 2\sqrt{x} })\]

Answer 6

\[\frac{ 1 }{ 6x} *(2+\sqrt{x})^3 \]

Answer 7

@imqwerty

Answer 8

can you help me here my answer is probably wrong,i think

Answer 9

yeah its wrong
you did \(u\) substitution where \(u=2+\sqrt{x}\)
right?

Answer 10

so what will happens here?

Answer 11

1/3(2++sqrt of x)^3(1/2sqrtx) ?

Answer 12

1/6sqrtx(2+sqrtx)^3 +c ?

Answer 13

its not correct :)

Answer 14

we have this -> \(\color{blueviolet}{u=2+\sqrt{x}}\)

now we find the value of \(dx\)

\(u=2+\sqrt{x}\)
\(\color{blueviolet}{du=\large \frac{1}{2\sqrt{x}} dx}\)

when we have this- \(\Large \int \frac{(2+\sqrt{x})^3}{\sqrt{x}}dx\)

substituting the purple values we get this- \(\large \int 2u^3 du\)

try continuing from here :)

Answer 15

-2(2+sqrtx)^3(1/2qrtx)

Answer 16

-1/sqrtx (2+sqrtx)^3 +c ?

Answer 17

2/3 (u^3) +c ?

Answer 18

opss

Answer 19

2/6sqrtx (u^3)+c

Answer 20

where did you get the 2 in 2u^2du ?

Answer 21

no this is not correct
and i made a lil typo out there

it must be this-\(\Large \int \frac{(2+\sqrt{x})^{\color{red}{2}}}{\sqrt{x}}dx\)
=\(\large \int 2u^{\color{red}2} du\)

and then after we integrate we get this-
\(2 \Large \frac{u^{2+1}}{2+1}+c\)

sorry the site is lagging a bit for me

Answer 22

and yeah \(\large \frac{2}{3} u^3 +c\) was correct

Answer 23

the \(2\) from \(\large\color{blue}{2}u^3 du\) goes here(in blue)-> \(\color{blue}{2}\large \frac{u^{2+1}}{2+1}+c\)

Answer 24

where did you get that 2?

Answer 25

how did you arrive to get that?

Answer 26

okay
we had this-> \[\int\limits_{}^{}2u^3 du\]here that 2 is a constant so we take it out we get this=\[2\int\limits_{}^{}u^3 du\]now we integrate the inner part and get this->\(2 \Large \frac{u^{2+1}}{2+1}+c\)

Answer 27

^^ thanks again

Answer 28

np :)
now just substitute the value of \(u\) to get the final answer