What is the integral from 1 to e^6 of x^3ln(x) ?

The integral from 1 to e^6 of x^3ln(x) is (23e^{24}+1)/(16)

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integral from 1 to e^6 of x^3ln(x)

\int_{1}^{e^{6}} x^{3} ln (x) d x

\frac{23 e ^{24} + 1}{16}

Decimal

38078113061.71244

Solution steps

\int_{1}^{e^{6}} x^{3} ln (x) d x

Apply Integration By Parts

= [\frac{1}{4} (x^{4} ln (x) - 4 \cdot \int \frac{x ^{3}}{4} d x)]_{1}^{e^{6}}

\int \frac{x ^{3}}{4} d x = \frac{x ^{4}}{16}

= [\frac{1}{4} (x^{4} ln (x) - 4 \cdot \frac{x ^{4}}{16})]_{1}^{e^{6}}

4 \cdot \frac{x ^{4}}{16} = \frac{x ^{4}}{4}

= [\frac{1}{4} (x^{4} ln (x) - \frac{x ^{4}}{4})]_{1}^{e^{6}}

Compute the boundaries:

\frac{23 e ^{24}}{16} + \frac{1}{16}

= \frac{23 e ^{24}}{16} + \frac{1}{16}

Simplify

= \frac{23 e ^{24} + 1}{16}

What is the integral from 1 to e^6 of x^3ln(x) ?
The integral from 1 to e^6 of x^3ln(x) is (23e^{24}+1)/(16)