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

The integral from 1 to 3 of x/((2x-1)^3) is 8/25

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

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

\frac{8}{25}

Decimal Notation

0.32

Solution steps

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

Take the partial fraction of

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int_{1}^{3} \frac{1}{2 ( 2 x - 1 ) ^{3}} d x = \frac{3}{25}

= \frac{1}{5} + \frac{3}{25}

Simplify

\frac{1}{5} + \frac{3}{25} : \frac{8}{25}

= \frac{8}{25}

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What is the integral from 1 to 3 of x/((2x-1)^3) ?
The integral from 1 to 3 of x/((2x-1)^3) is 8/25