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

The integral from 1 to e of x^4ln(x) is (4e^5+1)/(25)

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

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

\frac{4 e ^{5} + 1}{25}

Decimal Notation

23.78610 \dots

Solution steps

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

Apply Integration By Parts

: [\frac{1}{5} (x^{5} ln (x) - 5 \cdot \int \frac{x ^{4}}{5} d x)]_{1}^{e}

= [\frac{1}{5} (x^{5} ln (x) - 5 \cdot \int \frac{x ^{4}}{5} d x)]_{1}^{e}

\int \frac{x ^{4}}{5} d x = \frac{x ^{5}}{25}

= [\frac{1}{5} (x^{5} ln (x) - 5 \cdot \frac{x ^{5}}{25})]_{1}^{e}

5 \cdot \frac{x ^{5}}{25} = \frac{x ^{5}}{5}

= [\frac{1}{5} (x^{5} ln (x) - \frac{x ^{5}}{5})]_{1}^{e}

Compute the boundaries:

\frac{4 e ^{5}}{25} + \frac{1}{25}

= \frac{4 e ^{5}}{25} + \frac{1}{25}

Simplify

= \frac{4 e ^{5} + 1}{25}

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What is the integral from 1 to e of x^4ln(x) ?
The integral from 1 to e of x^4ln(x) is (4e^5+1)/(25)