What is the integral of ((ln(53))/v-ln(6)v^e) ?

The integral of ((ln(53))/v-ln(6)v^e) is ln(53)ln|v|-1/(e+1)ln(6)v^{e+1}+C

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integral of ((ln(53))/v-ln(6)v^e)

\int (\frac{ln ( 53 )}{v} - ln (6) v^{e}) d v

ln (53) ln ∣ v ∣ - \frac{1}{e + 1} ln (6) v^{e + 1} + C

Solution steps

\int \frac{ln ( 53 )}{v} - ln (6) v^{e} d v

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{ln ( 53 )}{v} d v - \int ln (6) v^{e} d v

\int \frac{ln ( 53 )}{v} d v = ln (53) ln ∣ v ∣

\int ln (6) v^{e} d v = \frac{1}{e + 1} ln (6) v^{e + 1}

= ln (53) ln ∣ v ∣ - \frac{1}{e + 1} ln (6) v^{e + 1}

Add a constant to the solution

= ln (53) ln ∣ v ∣ - \frac{1}{e + 1} ln (6) v^{e + 1} + C

What is the integral of ((ln(53))/v-ln(6)v^e) ?
The integral of ((ln(53))/v-ln(6)v^e) is ln(53)ln|v|-1/(e+1)ln(6)v^{e+1}+C