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

The integral of (x+1)/(xsqrt(x-4)) is 2sqrt(x-4)+arctan(1/2 sqrt(x-4))+C

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integral of (x+1)/(xsqrt(x-4))

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

What is the integral of (x+1)/(xsqrt(x-4)) ?
The integral of (x+1)/(xsqrt(x-4)) is 2sqrt(x-4)+arctan(1/2 sqrt(x-4))+C