integral from 1 to 4 of (1+ln(x))x^x

解答

\int_{1}^{4} (1 + ln (x)) x^{x} d x

255

求解步骤

\int_{1}^{4} (1 + ln (x)) x^{x} d x

乘以

1 + ln (x)

的共轭根式

: \frac{1 - ln ^{2} ( x )}{1 - ln ( x )}

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

找到无定义的点（奇点）:

x = e

如果存在

b, a < b < c, f (b) =

无定义

, \int_{a}^{c} f (x) d x = \int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x

= \int_{1}^{e} \frac{1 - ln ^{2} ( x )}{1 - ln ( x )} x^{x} d x + \int_{e}^{4} \frac{1 - ln ^{2} ( x )}{1 - ln ( x )} x^{x} d x

\int_{1}^{e} \frac{1 - ln ^{2} ( x )}{1 - ln ( x )} x^{x} d x = e^{e} - 1

\int_{e}^{4} \frac{1 - ln ^{2} ( x )}{1 - ln ( x )} x^{x} d x = 256 - e^{e}

= e^{e} - 1 + 256 - e^{e}

化简

= 255

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