What is the integral from 1 to 5 of x/(sqrt(2x-3)) ?

The integral from 1 to 5 of x/(sqrt(2x-3)) is (8sqrt(7))/3-i 4/3

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integral from 1 to 5 of x/(sqrt(2x-3))

\int_{1}^{5} \frac{x}{2 x - 3} d x

\frac{8 7}{3} - i \frac{4}{3}

Solution steps

\int_{1}^{5} \frac{x}{2 x - 3} d x

Find undefined (singularity) points:

x = \frac{3}{2}

If exist

b, a < b < c, f (b) =

undefined

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{1}^{\frac{3}{2}} \frac{x}{2 x - 3} d x + \int_{\frac{3}{2}}^{5} \frac{x}{2 x - 3} d x

\int_{1}^{\frac{3}{2}} \frac{x}{2 x - 3} d x = - \frac{4 i}{3}

\int_{\frac{3}{2}}^{5} \frac{x}{2 x - 3} d x = \frac{8 7}{3}

= - \frac{4 i}{3} + \frac{8 7}{3}

Simplify

- \frac{4 i}{3} + \frac{8 7}{3} : \frac{8 7}{3} - i \frac{4}{3}

= \frac{8 7}{3} - i \frac{4}{3}

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What is the integral from 1 to 5 of x/(sqrt(2x-3)) ?
The integral from 1 to 5 of x/(sqrt(2x-3)) is (8sqrt(7))/3-i 4/3