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

The integral from 3/2 to 2 of xsqrt(2x-3) is 3/5

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integral from 3/2 to 2 of xsqrt(2x-3)

\int_{\frac{3}{2}}^{2} x 2 x - 3 d x

\frac{3}{5}

Solution steps

\int_{\frac{3}{2}}^{2} x 2 x - 3 d x

Apply u-substitution

: \int_{0}^{1} \frac{u ( u + 3 )}{4} d u

= \int_{0}^{1} \frac{u ( u + 3 )}{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_{0}^{1} u (u + 3) d u

Expand

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

= \frac{1}{4} \cdot \int_{0}^{1} u^{\frac{3}{2}} + 3 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_{0}^{1} u^{\frac{3}{2}} d u + \int_{0}^{1} 3 u d u)

\int_{0}^{1} u^{\frac{3}{2}} d u = \frac{2}{5}

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

= \frac{1}{4} (\frac{2}{5} + 2)

Simplify

\frac{1}{4} (\frac{2}{5} + 2) : \frac{3}{5}

= \frac{3}{5}

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What is the integral from 3/2 to 2 of xsqrt(2x-3) ?
The integral from 3/2 to 2 of xsqrt(2x-3) is 3/5