Question

x/√x+1,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

Dear Student,

Solution:

To integrate such functions, we first convert the function into that form which can be integrated

$f(x) = \frac{x}{ \sqrt{x + 1} }$
So, put
$x + 1 = {t}^{2}$
now differentiate both side
$dx = 2t \: dt \\ \\ x = {t}^{2} - 1$
please assume sign of integration in front of function,

substitute the assumption
$2t(\frac{ {t}^{2} - 1 }{t}) \: dt \\ \\ =integrate \: \: 2[\frac{ {t}^{3} }{t} \: dt - integrate \: \: \frac{t}{t} \: dt] \\ \\ = \frac{ {t}^{3} }{3} - t+ c$
undo substitute

final answer is in attachment,due to some reason it is not shown exactly.

Answer 2

HELLO DEAR,


GIVEN:-

∫x/√(x + 1).dx

put (x + 1) = t² also, x = t² - 1
=> dx = 2t.dt

therefore, $\bold{I = 2\int(t^2 - 1)\,dt}$

$\bold{I = 2(t^3/3 - t) + C}$

put the value of t = √(x + 1)

$\bold{I = 2[(x + 1)^{3/2}/3 - \sqrt{x + 1} ] + C}$

$\bold{I = 2/3(x + 1)^{3/2} - 2(x + 1)^{1/2} + C.}$


I HOPE ITS HELP YOU DEAR,
THANKS