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

The integral of (sqrt(x)log_{10}(2x)) is 2/3 x^{3/2}log_{10}(2x)-(4x^{3/2})/(9ln(10))+C

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

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

\frac{2}{3} x^{\frac{3}{2}} lo g_{10} (2 x) - \frac{4 x ^{\frac{3}{2}}}{9 ln ( 10 )} + C

Solution steps

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

Apply Integration By Parts

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

\int \frac{2 x ^{\frac{1}{2}}}{3 ln ( 10 )} d x = \frac{4 x ^{\frac{3}{2}}}{9 ln ( 10 )}

= \frac{2}{3} x^{\frac{3}{2}} lo g_{10} (2 x) - \frac{4 x ^{\frac{3}{2}}}{9 ln ( 10 )}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} lo g_{10} (2 x) - \frac{4 x ^{\frac{3}{2}}}{9 ln ( 10 )} + C

What is the integral of (sqrt(x)log_{10}(2x)) ?
The integral of (sqrt(x)log_{10}(2x)) is 2/3 x^{3/2}log_{10}(2x)-(4x^{3/2})/(9ln(10))+C