What is the integral of log_{5}(2x-1) ?

The integral of log_{5}(2x-1) is xlog_{5}(2x-1)-(2x-1+ln|2x-1|)/(2ln(5))+C

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integral of log_{5}(2x-1)

\int lo g_{5} (2 x - 1) d x

x lo g_{5} (2 x - 1) - \frac{2 x - 1 + ln ∣ 2 x - 1 ∣}{2 ln ( 5 )} + C

Solution steps

\int lo g_{5} (2 x - 1) d x

Apply Integration By Parts

: x lo g_{5} (2 x - 1) - \int \frac{2 x}{ln ( 5 ) ( 2 x - 1 )} d x

= x lo g_{5} (2 x - 1) - \int \frac{2 x}{ln ( 5 ) ( 2 x - 1 )} d x

\int \frac{2 x}{ln ( 5 ) ( 2 x - 1 )} d x = \frac{2 x - 1 + ln ∣ 2 x - 1 ∣}{2 ln ( 5 )}

= x lo g_{5} (2 x - 1) - \frac{2 x - 1 + ln ∣ 2 x - 1 ∣}{2 ln ( 5 )}

Add a constant to the solution

= x lo g_{5} (2 x - 1) - \frac{2 x - 1 + ln ∣ 2 x - 1 ∣}{2 ln ( 5 )} + C

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What is the integral of log_{5}(2x-1) ?
The integral of log_{5}(2x-1) is xlog_{5}(2x-1)-(2x-1+ln|2x-1|)/(2ln(5))+C