What is the integral of (x+3)/(sqrt(x-1)) ?

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

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integral of (x+3)/(sqrt(x-1))

\int \frac{x + 3}{x - 1} d x

\frac{2}{3} (x - 1)^{\frac{3}{2}} + 8 x - 1 + C

Solution steps

\int \frac{x + 3}{x - 1} d x

Expand

\frac{x + 3}{x - 1} : \frac{x}{x - 1} + \frac{3}{x - 1}

= \int \frac{x}{x - 1} + \frac{3}{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{x}{x - 1} d x + \int \frac{3}{x - 1} d x

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

\int \frac{3}{x - 1} d x = 6 x - 1

= 2 (\frac{1}{3} (x - 1)^{\frac{3}{2}} + x - 1) + 6 x - 1

Simplify

2 (\frac{1}{3} (x - 1)^{\frac{3}{2}} + x - 1) + 6 x - 1 : \frac{2}{3} (x - 1)^{\frac{3}{2}} + 8 x - 1

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

Add a constant to the solution

= \frac{2}{3} (x - 1)^{\frac{3}{2}} + 8 x - 1 + C

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