What is the integral of log_{10}(x+5) ?

The integral of log_{10}(x+5) is xlog_{10}(x+5)-1/(ln(10))(x+5-5ln|x+5|)+C

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integral of log_{10}(x+5)

\int lo g_{10} (x + 5) d x

x lo g_{10} (x + 5) - \frac{1}{ln ( 10 )} (x + 5 - 5 ln ∣ x + 5 ∣) + C

Solution steps

\int lo g_{10} (x + 5) d x

Apply Integration By Parts

: x lo g_{10} (x + 5) - \int \frac{x}{ln ( 10 ) ( x + 5 )} d x

= x lo g_{10} (x + 5) - \int \frac{x}{ln ( 10 ) ( x + 5 )} d x

\int \frac{x}{ln ( 10 ) ( x + 5 )} d x = \frac{1}{ln ( 10 )} (x + 5 - 5 ln ∣ x + 5 ∣)

= x lo g_{10} (x + 5) - \frac{1}{ln ( 10 )} (x + 5 - 5 ln ∣ x + 5 ∣)

Add a constant to the solution

= x lo g_{10} (x + 5) - \frac{1}{ln ( 10 )} (x + 5 - 5 ln ∣ x + 5 ∣) + C

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What is the integral of log_{10}(x+5) ?
The integral of log_{10}(x+5) is xlog_{10}(x+5)-1/(ln(10))(x+5-5ln|x+5|)+C