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

解

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

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

解答ステップ

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

部分積分を適用する

= [\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}

限度を計算する:

\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 )}

簡素化

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