What is the integral of (8x+4)/(sqrt(4x-1)) ?

The integral of (8x+4)/(sqrt(4x-1)) is 1/3 (4x-1)^{3/2}+3sqrt(4x-1)+C

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integral of (8x+4)/(sqrt(4x-1))

\int \frac{8 x + 4}{4 x - 1} d x

\frac{1}{3} (4 x - 1)^{\frac{3}{2}} + 3 4 x - 1 + C

Solution steps

\int \frac{8 x + 4}{4 x - 1} d x

Expand

\frac{8 x + 4}{4 x - 1} : \frac{8 x}{4 x - 1} + \frac{4}{4 x - 1}

= \int \frac{8 x}{4 x - 1} + \frac{4}{4 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 \frac{8 x}{4 x - 1} d x + \int \frac{4}{4 x - 1} d x

\int \frac{8 x}{4 x - 1} d x = \frac{1}{3} (4 x - 1)^{\frac{3}{2}} + 4 x - 1

\int \frac{4}{4 x - 1} d x = 2 4 x - 1

= \frac{1}{3} (4 x - 1)^{\frac{3}{2}} + 4 x - 1 + 2 4 x - 1

Simplify

= \frac{1}{3} (4 x - 1)^{\frac{3}{2}} + 3 4 x - 1

Add a constant to the solution

= \frac{1}{3} (4 x - 1)^{\frac{3}{2}} + 3 4 x - 1 + C

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What is the integral of (8x+4)/(sqrt(4x-1)) ?
The integral of (8x+4)/(sqrt(4x-1)) is 1/3 (4x-1)^{3/2}+3sqrt(4x-1)+C