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

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

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

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

\frac{1}{3}

Decimal

0.33333 \dots

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}

Compute the boundaries:

\frac{2}{3}

= \frac{1}{2} \cdot \frac{2}{3}

Simplify

= \frac{1}{3}

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