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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

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