What is the integral from 0 to 2 of x^7ln(x) ?

The integral from 0 to 2 of x^7ln(x) is 32ln(2)-4

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integral from 0 to 2 of x^7ln(x)

\int_{0}^{2} x^{7} ln (x) d x

32 ln (2) - 4

Decimal

18.18070 \dots

Solution steps

\int_{0}^{2} x^{7} ln (x) d x

Apply Integration By Parts

: [\frac{1}{8} x^{8} ln (x) - \int \frac{x ^{7}}{8} d x]_{0}^{2}

= [\frac{1}{8} x^{8} ln (x) - \int \frac{x ^{7}}{8} d x]_{0}^{2}

\int \frac{x ^{7}}{8} d x = \frac{x ^{8}}{64}

= [\frac{1}{8} x^{8} ln (x) - \frac{x ^{8}}{64}]_{0}^{2}

Compute the boundaries:

32 ln (2) - 4

= 32 ln (2) - 4

\int_{0}^{2} r^{3} 1 + 4 r^{2} d r

\int_{- 1}^{1} \frac{1}{e ^{5 t}} d t

\int_{1}^{3} 7 x^{2} d x

\int_{- 3}^{3} x^{2} - 4 d x

\int_{2}^{5} (x^{3} + 2) d x

What is the integral from 0 to 2 of x^7ln(x) ?
The integral from 0 to 2 of x^7ln(x) is 32ln(2)-4