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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

\int \frac{7}{3} d x

\frac{1}{x}, at x = 4

\frac{d}{d x} (cot (9) (1 - 2 x^{2}))

n \to 0 lim (n sin (\frac{1}{n}))

\int ln (x^{2} - 4) d x

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