What is the integral of (log_{5}(t))/t ?

The integral of (log_{5}(t))/t is log_{5}(t)ln|t|-(ln|t|^2)/(2ln(5))+C

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integral of (log_{5}(t))/t

\int \frac{lo g _{5} ( t )}{t} d t

lo g_{5} (t) ln ∣ t ∣ - \frac{ln ∣ t ∣ ^{2}}{2 ln ( 5 )} + C

Solution steps

\int \frac{lo g _{5} ( t )}{t} d t

Apply Integration By Parts

: lo g_{5} (t) ln ∣ t ∣ - \int \frac{ln ∣ t ∣}{ln ( 5 ) t} d t

= lo g_{5} (t) ln ∣ t ∣ - \int \frac{ln ∣ t ∣}{ln ( 5 ) t} d t

\int \frac{ln ∣ t ∣}{ln ( 5 ) t} d t = \frac{ln ∣ t ∣ ^{2}}{2 ln ( 5 )}

= lo g_{5} (t) ln ∣ t ∣ - \frac{ln ∣ t ∣ ^{2}}{2 ln ( 5 )}

Add a constant to the solution

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

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