integral of 3x(ln(x^7))^2

解答

\int 3 x (ln (x^{7}))^{2} d x

3 (\frac{49}{2} x^{2} ln^{2} (x) - 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4})) + C

求解步骤

\int 3 x (ln (x^{7}))^{2} d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int x (ln (x^{7}))^{2} d x

使用分布积分法

= 3 (\frac{1}{2} x^{2} ln^{2} (x^{7}) - \int 7 x ln (x^{7}) d x)

\int 7 x ln (x^{7}) d x = 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4})

= 3 (\frac{1}{2} x^{2} ln^{2} (x^{7}) - 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4}))

\frac{1}{2} x^{2} ln^{2} (x^{7}) = \frac{49}{2} x^{2} ln^{2} (x)

= 3 (\frac{49}{2} x^{2} ln^{2} (x) - 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4}))

解答补常数

= 3 (\frac{49}{2} x^{2} ln^{2} (x) - 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4})) + C

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