What is the integral from 1 to 4 of (1+ln(x))x^x ?

The integral from 1 to 4 of (1+ln(x))x^x is 255

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

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

255

Solution steps

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

Multiply by the conjugate of

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

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

Find undefined (singularity) points:

x = e

If exist

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

undefined

, \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}

Simplify

= 255

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What is the integral from 1 to 4 of (1+ln(x))x^x ?
The integral from 1 to 4 of (1+ln(x))x^x is 255