integral of xln^5(x)

解

\int x ln^{5} (x) d x

\frac{1}{2} x^{2} ln^{5} (x) - \frac{5}{8} (2 x^{2} ln^{4} (x) - 4 x^{2} ln^{3} (x) + 6 x^{2} ln^{2} (x) - 6 x^{2} ln (x) + 3 x^{2}) + C

解答ステップ

\int x ln^{5} (x) d x

部分積分を適用する

= \frac{1}{2} x^{2} ln^{5} (x) - \int \frac{5}{2} x ln^{4} (x) d x

\int \frac{5}{2} x ln^{4} (x) d x = \frac{5}{8} (2 x^{2} ln^{4} (x) - 4 x^{2} ln^{3} (x) + 6 x^{2} ln^{2} (x) - 6 x^{2} ln (x) + 3 x^{2})

= \frac{1}{2} x^{2} ln^{5} (x) - \frac{5}{8} (2 x^{2} ln^{4} (x) - 4 x^{2} ln^{3} (x) + 6 x^{2} ln^{2} (x) - 6 x^{2} ln (x) + 3 x^{2})

定数を解答に追加する

= \frac{1}{2} x^{2} ln^{5} (x) - \frac{5}{8} (2 x^{2} ln^{4} (x) - 4 x^{2} ln^{3} (x) + 6 x^{2} ln^{2} (x) - 6 x^{2} ln (x) + 3 x^{2}) + C