What is the integral of x^2log_{10}(2x) ?

The integral of x^2log_{10}(2x) is 1/3 x^3log_{10}(2x)-(x^3)/(9ln(10))+C

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integral of x^2log_{10}(2x)

\int x^{2} lo g_{10} (2 x) d x

\frac{1}{3} x^{3} lo g_{10} (2 x) - \frac{x ^{3}}{9 ln ( 10 )} + C

Solution steps

\int x^{2} lo g_{10} (2 x) d x

Apply Integration By Parts

: \frac{1}{3} x^{3} lo g_{10} (2 x) - \int \frac{x ^{2}}{3 ln ( 10 )} d x

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

\int \frac{x ^{2}}{3 ln ( 10 )} d x = \frac{x ^{3}}{9 ln ( 10 )}

= \frac{1}{3} x^{3} lo g_{10} (2 x) - \frac{x ^{3}}{9 ln ( 10 )}

Add a constant to the solution

= \frac{1}{3} x^{3} lo g_{10} (2 x) - \frac{x ^{3}}{9 ln ( 10 )} + C

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What is the integral of x^2log_{10}(2x) ?
The integral of x^2log_{10}(2x) is 1/3 x^3log_{10}(2x)-(x^3)/(9ln(10))+C