What is the integral of (5x-2)/((1-2x)^2(3-x)) ?

The integral of (5x-2)/((1-2x)^2(3-x)) is 13/25 ln|2x-1|-1/(10(2x-1))-13/25 ln|x-3|+C

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integral of (5x-2)/((1-2x)^2(3-x))

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

\frac{13}{25} ln ∣ 2 x - 1 ∣ - \frac{1}{10 ( 2 x - 1 )} - \frac{13}{25} ln ∣ x - 3 ∣ + C

Solution steps

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

Take the partial fraction of

\frac{5 x - 2}{( 1 - 2 x ) ^{2} ( 3 - x )} : \frac{26}{25 ( 2 x - 1 )} + \frac{1}{5 ( 2 x - 1 ) ^{2}} - \frac{13}{25 ( x - 3 )}

= \int \frac{26}{25 ( 2 x - 1 )} + \frac{1}{5 ( 2 x - 1 ) ^{2}} - \frac{13}{25 ( x - 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 \frac{26}{25 ( 2 x - 1 )} d x + \int \frac{1}{5 ( 2 x - 1 ) ^{2}} d x - \int \frac{13}{25 ( x - 3 )} d x

\int \frac{26}{25 ( 2 x - 1 )} d x = \frac{13}{25} ln ∣ 2 x - 1 ∣

\int \frac{1}{5 ( 2 x - 1 ) ^{2}} d x = - \frac{1}{10 ( 2 x - 1 )}

\int \frac{13}{25 ( x - 3 )} d x = \frac{13}{25} ln ∣ x - 3 ∣

= \frac{13}{25} ln ∣ 2 x - 1 ∣ - \frac{1}{10 ( 2 x - 1 )} - \frac{13}{25} ln ∣ x - 3 ∣

Add a constant to the solution

= \frac{13}{25} ln ∣ 2 x - 1 ∣ - \frac{1}{10 ( 2 x - 1 )} - \frac{13}{25} ln ∣ x - 3 ∣ + C

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What is the integral of (5x-2)/((1-2x)^2(3-x)) ?
The integral of (5x-2)/((1-2x)^2(3-x)) is 13/25 ln|2x-1|-1/(10(2x-1))-13/25 ln|x-3|+C