What is the integral of ((x+5))/(sqrt(x^2-8x)) ?

The integral of ((x+5))/(sqrt(x^2-8x)) is sqrt(x^2-8x)+9ln(1/4 |sqrt(x^2-8x)+x-4|)+C

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integral of ((x+5))/(sqrt(x^2-8x))

\int \frac{( x + 5 )}{x ^{2} - 8 x} d x

x^{2} - 8 x + 9 ln (\frac{1}{4} x^{2} - 8 x + x - 4) + C

Solution steps

\int \frac{x + 5}{x ^{2} - 8 x} d x

Expand

\frac{x + 5}{x ^{2} - 8 x} : \frac{x}{x ^{2} - 8 x} + \frac{5}{x ^{2} - 8 x}

= \int \frac{x}{x ^{2} - 8 x} + \frac{5}{x ^{2} - 8 x} 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 ^{2} - 8 x} d x + \int \frac{5}{x ^{2} - 8 x} d x

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

\int \frac{5}{x ^{2} - 8 x} d x = 5 ln (\frac{1}{4} x^{2} - 8 x + x - 4)

= x^{2} - 8 x + 4 ln (\frac{1}{4} x^{2} - 8 x + x - 4) + 5 ln (\frac{1}{4} x^{2} - 8 x + x - 4)

Simplify

= x^{2} - 8 x + 9 ln (\frac{1}{4} x^{2} - 8 x + x - 4)

Add a constant to the solution

= x^{2} - 8 x + 9 ln (\frac{1}{4} x^{2} - 8 x + x - 4) + C

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What is the integral of ((x+5))/(sqrt(x^2-8x)) ?
The integral of ((x+5))/(sqrt(x^2-8x)) is sqrt(x^2-8x)+9ln(1/4 |sqrt(x^2-8x)+x-4|)+C