What is the integral from 0 to x of ylog_{10}(y) ?

The integral from 0 to x of ylog_{10}(y) is (x^2log_{10}(x))/2-(x^2)/(4ln(10))

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integral from 0 to x of ylog_{10}(y)

\int_{0}^{x} y lo g_{10} (y) d y

\frac{x ^{2} lo g _{10} ( x )}{2} - \frac{x ^{2}}{4 ln ( 10 )}

Solution steps

\int_{0}^{x} y lo g_{10} (y) d y

Apply Integration By Parts

= [\frac{1}{2} y^{2} lo g_{10} (y) - \int \frac{y}{2 ln ( 10 )} d y]_{0}^{x}

\int \frac{y}{2 ln ( 10 )} d y = \frac{y ^{2}}{4 ln ( 10 )}

= [\frac{1}{2} y^{2} lo g_{10} (y) - \frac{y ^{2}}{4 ln ( 10 )}]_{0}^{x}

Compute the boundaries:

\frac{1}{2} x^{2} lo g_{10} (x) - \frac{x ^{2}}{4 ln ( 10 )}

= \frac{1}{2} x^{2} lo g_{10} (x) - \frac{x ^{2}}{4 ln ( 10 )}

Simplify

= \frac{x ^{2} lo g _{10} ( x )}{2} - \frac{x ^{2}}{4 ln ( 10 )}

What is the integral from 0 to x of ylog_{10}(y) ?
The integral from 0 to x of ylog_{10}(y) is (x^2log_{10}(x))/2-(x^2)/(4ln(10))