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

2 e^{x} (cos (x) - sin (x))

\int_{- 2}^{3} (x^{3} - x^{2} - 6 x) d x

\frac{d}{d t} (arcsin (t))

\frac{d}{d t} (cos (t) + sin (t))

y = x^{2} - 19, y = 18 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