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

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

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

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

\frac{π}{2}

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

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

= 2 [arctan (u)]_{0}^{1}

Compute the boundaries:

\frac{π}{4}

= 2 \cdot \frac{π}{4}

Simplify

= \frac{π}{2}

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