What is the integral from 1/2 to 1 of x^4ln(2x) ?

The integral from 1/2 to 1 of x^4ln(2x) is -31/800+1/5 ln(2)

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integral from 1/2 to 1 of x^4ln(2x)

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

- \frac{31}{800} + \frac{1}{5} ln (2)

Solution steps

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

Apply Integration By Parts

: [\frac{1}{5} x^{5} ln (2 x) - \int \frac{x ^{4}}{5} d x]_{\frac{1}{2}}^{1}

= [\frac{1}{5} x^{5} ln (2 x) - \int \frac{x ^{4}}{5} d x]_{\frac{1}{2}}^{1}

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

= [\frac{1}{5} x^{5} ln (2 x) - \frac{x ^{5}}{25}]_{\frac{1}{2}}^{1}

Compute the boundaries:

- \frac{31}{800} + \frac{1}{5} ln (2)

= - \frac{31}{800} + \frac{1}{5} ln (2)

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

What is the integral from 1/2 to 1 of x^4ln(2x) ?
The integral from 1/2 to 1 of x^4ln(2x) is -31/800+1/5 ln(2)