integral of x^7ln(x^4)

解答

\int x^{7} ln (x^{4}) d x

4 (\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64}) + C

求解步骤

\int x^{7} ln (x^{4}) d x

化简

ln (x^{4}) : 4 ln (x)

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

提出常数:

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

= 4 \cdot \int x^{7} ln (x) d x

使用分布积分法

= 4 (\frac{1}{8} x^{8} ln (x) - \int \frac{x ^{7}}{8} d x)

\int \frac{x ^{7}}{8} d x = \frac{x ^{8}}{64}

= 4 (\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64})

解答补常数

= 4 (\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64}) + C

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