What is the integral from 1 to 3 of (sqrt(x^2-1))/x ?

The integral from 1 to 3 of (sqrt(x^2-1))/x is -arctan(2sqrt(2))+2sqrt(2)

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integral from 1 to 3 of (sqrt(x^2-1))/x

\int_{1}^{3} \frac{x ^{2} - 1}{x} d x

- arctan (22) + 22

Decimal Notation

1.59746 \dots

Solution steps

\int_{1}^{3} \frac{x ^{2} - 1}{x} d x

Apply u-substitution

: \int_{0}^{22} \frac{u ^{2}}{u ^{2} + 1} d u

= \int_{0}^{22} \frac{u ^{2}}{u ^{2} + 1} d u

\frac{u ^{2}}{u ^{2} + 1} = - \frac{1}{u ^{2} + 1} + 1

= \int_{0}^{22} - \frac{1}{u ^{2} + 1} + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int_{0}^{22} \frac{1}{u ^{2} + 1} d u + \int_{0}^{22} 1 d u

\int_{0}^{22} \frac{1}{u ^{2} + 1} d u = arctan (22)

\int_{0}^{22} 1 d u = 22

= - arctan (22) + 22

\int arctan (x - 1) d x

\int \frac{1}{( x ^{3} - 1 )} d x

\int \frac{2 x ^{2} - 1}{x ( x + 1 ) ^{3}} d x

\frac{d}{d x} (tan (1 - x))

\frac{\partial}{\partial x} (x^{2} + 10 x y + y^{2})

What is the integral from 1 to 3 of (sqrt(x^2-1))/x ?
The integral from 1 to 3 of (sqrt(x^2-1))/x is -arctan(2sqrt(2))+2sqrt(2)