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

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

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

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

- \frac{π}{3}

Decimal

- 1.04719 \dots

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= - [arctan (u)]_{0}^{3}

Compute the boundaries:

\frac{π}{3}

= - \frac{π}{3}

\int_{- 1}^{1} t^{2} d t

\int_{0}^{t} cos (x) d x

\int_{0}^{π} cos^{2} (\frac{x}{2}) d x

\int_{0}^{1} a x + b x^{2} d x

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

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