Question

1/√5-3x,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

In the attachment I have answered this problem.         The solution is simple and easy to understand.             See the attachment for detailed solution.

Answer 2

Dear Student,

Answer:
$- \frac{2}{3} \sqrt{(5 - 3x)} + c$


Solution:

To integrate
$\frac{1}{ \sqrt{5 - 3x} }$
w.r.t. x

let
$5 - 3x = t \\ \\ - 3dx = dt \\ \\ dx = \frac{ - dt}{3} \\ \\ = - \frac{1}{3 \sqrt{t} } dt$
as we know the power rule

${x}^{n} dx = \frac{ {x}^{n + 1} }{n + 1} \\ \\ - \frac{1}{3} \frac{ {t}^{ \frac{ - 1}{2} + 1 } }{ \frac{ - 1}{2} + 1 } + c \\ \\ = - \frac{2}{3} {t}^{ \frac{1}{2} } + c$
undo substitution
$= - \frac{2}{3} ( {5 - 3x)}^{ \frac{1}{2} } + c \\ \\ = - \frac{2}{3} \sqrt{(5 - 3x)} + c$

Hope it helps you.