What is the integral of (log_{10}(x^3))/x ?

The integral of (log_{10}(x^3))/x is 3log_{10}(x)-1/(ln(10))(3ln|x|-3/2 ln^2(x))+C

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integral of (log_{10}(x^3))/x

\int \frac{lo g _{10} ( x ^{3} )}{x} d x

3 lo g_{10} (x) - \frac{1}{ln ( 10 )} (3 ln ∣ x ∣ - \frac{3}{2} ln^{2} (x)) + C

Solution steps

\int \frac{lo g _{10} ( x ^{3} )}{x} d x

Apply Integration By Parts

: lo g_{10} (x^{3}) - \int \frac{3 - ln ( x ^{3} )}{ln ( 10 ) x} d x

= lo g_{10} (x^{3}) - \int \frac{3 - ln ( x ^{3} )}{ln ( 10 ) x} d x

\int \frac{3 - ln ( x ^{3} )}{ln ( 10 ) x} d x = \frac{1}{ln ( 10 )} (3 ln ∣ x ∣ - \frac{3}{2} ln^{2} (x))

= lo g_{10} (x^{3}) - \frac{1}{ln ( 10 )} (3 ln ∣ x ∣ - \frac{3}{2} ln^{2} (x))

Apply log rule

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x),

assuming

x \geq 0

= 3 lo g_{10} (x) - \frac{1}{ln ( 10 )} (3 ln ∣ x ∣ - \frac{3}{2} ln^{2} (x))

Add a constant to the solution

= 3 lo g_{10} (x) - \frac{1}{ln ( 10 )} (3 ln ∣ x ∣ - \frac{3}{2} ln^{2} (x)) + C

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What is the integral of (log_{10}(x^3))/x ?
The integral of (log_{10}(x^3))/x is 3log_{10}(x)-1/(ln(10))(3ln|x|-3/2 ln^2(x))+C