What is the integral from e to e^2 of x^2ln(x) ?

The integral from e to e^2 of x^2ln(x) is (5e^6-2e^3)/9

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integral from e to e^2 of x^2ln(x)

\int_{e}^{e^{2}} x^{2} ln (x) d x

\frac{5 e ^{6} - 2 e ^{3}}{9}

Decimal Notation

219.66365 \dots

Solution steps

\int_{e}^{e^{2}} x^{2} ln (x) d x

Apply Integration By Parts

: [\frac{1}{3} (x^{3} ln (x) - 3 \cdot \int \frac{x ^{2}}{3} d x)]_{e}^{e^{2}}

= [\frac{1}{3} (x^{3} ln (x) - 3 \cdot \int \frac{x ^{2}}{3} d x)]_{e}^{e^{2}}

\int \frac{x ^{2}}{3} d x = \frac{x ^{3}}{9}

= [\frac{1}{3} (x^{3} ln (x) - 3 \cdot \frac{x ^{3}}{9})]_{e}^{e^{2}}

3 \cdot \frac{x ^{3}}{9} = \frac{x ^{3}}{3}

= [\frac{1}{3} (x^{3} ln (x) - \frac{x ^{3}}{3})]_{e}^{e^{2}}

Compute the boundaries:

\frac{5 e ^{6} - 2 e ^{3}}{9}

= \frac{5 e ^{6} - 2 e ^{3}}{9}

\int_{0}^{1} \frac{( y ^{2} + y ^{3} )}{2} d y

\int_{0}^{\frac{3}{2}} 3 - x^{2} d x

\int_{0}^{1 - x} x^{2} + y^{2} d y

\int_{2}^{4} - x^{3} + 7 x^{2} - 14 x + 8 d x

\int_{- 2}^{2} \frac{- 2}{( x - 1 ) ^{3}} d x

What is the integral from e to e^2 of x^2ln(x) ?
The integral from e to e^2 of x^2ln(x) is (5e^6-2e^3)/9