Question

integration of : ( dx / (root x(4x-25)))​

Answer 1

Question :

Integrate

$\sf\int\dfrac{dx}{\sqrt{4x-25}}$

Answer :

We have to Integrate

$\sf\int\dfrac{dx}{\sqrt{4x-25}}$

$\sf=\int\dfrac{dx}{\sqrt{(2x)^2-(5)^2}}$

Let, 2x = t then ,

$\sf\implies2dx=dt$

Then ,

$\sf\int\dfrac{dx}{\sqrt{(2x)^2-(5)^2}}$

$\sf=\dfrac{1}{2}\int\dfrac{dt}{\sqrt{(t)^2-(5)^2}}$

$\sf=\dfrac{1}{2}\int\dfrac{dt}{\sqrt{(t)^2-(5)^2}}$

$\sf=\dfrac{1}{2}\log(t+\sqrt{t^2-5^2})$

$\sf=\dfrac{1}{2}\log(2x+\sqrt{4x^2-25})$

Standard Formula's of Integration :

$1)\sf\int\dfrac{1}{\sqrt{(x)^2-(a)^2}}=\log(x+\sqrt{x^2-a^2})$

$2)\sf\int\dfrac{1}{\sqrt{(a)^2+(x)^2}}=\log(x+\sqrt{x^2+a^2})$