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

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

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

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

\frac{π}{3} - 3

Decimal Notation

- 0.68485 \dots

Solution steps

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

Apply u-substitution

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

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \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)

Simplify

= \frac{π}{3} - 3

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