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

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

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

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

- \frac{π}{3} + 3

Decimal

0.68485 \dots

Solution steps

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

Apply u-substitution

= \int_{0}^{3} \frac{u ^{2}}{u ^{2} + 1} d u

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

= \int_{0}^{3} - \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}^{3} \frac{1}{u ^{2} + 1} d u + \int_{0}^{3} 1 d u

\int_{0}^{3} \frac{1}{u ^{2} + 1} d u = \frac{π}{3}

\int_{0}^{3} 1 d u = 3

= - \frac{π}{3} + 3

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