What is the integral from 1 to 5 of xsqrt(2x-1) ?

The integral from 1 to 5 of xsqrt(2x-1) is 428/15

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integral from 1 to 5 of xsqrt(2x-1)

\int_{1}^{5} x 2 x - 1 d x

\frac{428}{15}

Decimal

28.53333 \dots

Solution steps

\int_{1}^{5} x 2 x - 1 d x

Apply u-substitution

= \int_{1}^{9} \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}^{9} u (u + 1) d u

Expand

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

= \frac{1}{4} \cdot \int_{1}^{9} 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}^{9} u^{\frac{3}{2}} d u + \int_{1}^{9} u d u)

\int_{1}^{9} u^{\frac{3}{2}} d u = \frac{484}{5}

\int_{1}^{9} u d u = \frac{52}{3}

= \frac{1}{4} (\frac{484}{5} + \frac{52}{3})

Simplify

\frac{1}{4} (\frac{484}{5} + \frac{52}{3}) : \frac{428}{15}

= \frac{428}{15}

What is the integral from 1 to 5 of xsqrt(2x-1) ?
The integral from 1 to 5 of xsqrt(2x-1) is 428/15