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

The integral from 1 to 2 of xsqrt(2x-1) is (21sqrt(3)-4)/(15)

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

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

\frac{21 3 - 4}{15}

Decimal Notation

2.15820 \dots

Solution steps

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

Apply u-substitution

: \int_{1}^{3} \frac{u ( u + 1 )}{4} d u

= \int_{1}^{3} \frac{u ( u + 1 )}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int_{1}^{3} u (u + 1) d u

Expand

u (u + 1) : u^{\frac{3}{2}} + u

= \frac{1}{4} \cdot \int_{1}^{3} u^{\frac{3}{2}} + u 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

= \frac{1}{4} (\int_{1}^{3} u^{\frac{3}{2}} d u + \int_{1}^{3} u d u)

\int_{1}^{3} u^{\frac{3}{2}} d u = \frac{18 3 - 2}{5}

\int_{1}^{3} u d u = 23 - \frac{2}{3}

= \frac{1}{4} (\frac{18 3 - 2}{5} + 23 - \frac{2}{3})

Simplify

\frac{1}{4} (\frac{18 3 - 2}{5} + 23 - \frac{2}{3}) : \frac{21 3 - 4}{15}

= \frac{21 3 - 4}{15}

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What is the integral from 1 to 2 of xsqrt(2x-1) ?
The integral from 1 to 2 of xsqrt(2x-1) is (21sqrt(3)-4)/(15)