What is the integral of (6x-1)/(sqrt(6x)-1) ?

The integral of (6x-1)/(sqrt(6x)-1) is (2sqrt(2))/(sqrt(3))x^{3/2}+x+C

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integral of (6x-1)/(sqrt(6x)-1)

\int \frac{6 x - 1}{6 x - 1} d x

\frac{2 2}{3} x^{\frac{3}{2}} + x + C

Solution steps

\int \frac{6 x - 1}{6 x - 1} d x

6 x - 1 = (6 x + 1) (6 x - 1)

= \int \frac{( 6 x + 1 ) ( 6 x - 1 )}{6 x - 1} d x

Simplify

\frac{( 6 x + 1 ) ( 6 x - 1 )}{6 x - 1} : 6 x + 1

= \int 6 x + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 6 x d x + \int 1 d x

\int 6 x d x = \frac{2 2}{3} x^{\frac{3}{2}}

\int 1 d x = x

= \frac{2 2}{3} x^{\frac{3}{2}} + x

Add a constant to the solution

= \frac{2 2}{3} x^{\frac{3}{2}} + x + C

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What is the integral of (6x-1)/(sqrt(6x)-1) ?
The integral of (6x-1)/(sqrt(6x)-1) is (2sqrt(2))/(sqrt(3))x^{3/2}+x+C