integral from 0 to 2x^2 of xln(x)

解

\int_{0}^{2 x^{2}} x ln (x) d x

2 x^{4} ln (2 x^{2}) - x^{4}

解答ステップ

\int_{0}^{2 x^{2}} x ln (x) d x

部分積分を適用する

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

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

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

限度を計算する:

2 x^{4} (ln (2) + 2 ln (x)) - x^{4}

= 2 x^{4} (ln (2) + 2 ln (x)) - x^{4}

簡素化

= 2 x^{4} ln (2 x^{2}) - x^{4}