What is the integral of (x+5)/(sqrt(x^2-1)) ?

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

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integral of (x+5)/(sqrt(x^2-1))

\int \frac{x + 5}{x ^{2} - 1} d x

x^{2} - 1 + 5 ln x^{2} - 1 + x + C

Solution steps

\int \frac{x + 5}{x ^{2} - 1} d x

Expand

\frac{x + 5}{x ^{2} - 1} : \frac{x}{x ^{2} - 1} + \frac{5}{x ^{2} - 1}

= \int \frac{x}{x ^{2} - 1} + \frac{5}{x ^{2} - 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 ^{2} - 1} d x + \int \frac{5}{x ^{2} - 1} d x

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

\int \frac{5}{x ^{2} - 1} d x = 5 ln x^{2} - 1 + x

= x^{2} - 1 + 5 ln x^{2} - 1 + x

Add a constant to the solution

= x^{2} - 1 + 5 ln x^{2} - 1 + x + C

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