What is the integral from 1/2 to 7 of 16xln(2x) ?

The integral from 1/2 to 7 of 16xln(2x) is -195+392ln(14)

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integral from 1/2 to 7 of 16xln(2x)

\int_{\frac{1}{2}}^{7} 16 x ln (2 x) d x

- 195 + 392 ln (14)

Decimal Notation

839.51047 \dots

Solution steps

\int_{\frac{1}{2}}^{7} 16 x ln (2 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 16 \cdot \int_{\frac{1}{2}}^{7} x ln (2 x) d x

Apply Integration By Parts

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

= 16 [\frac{1}{2} x^{2} ln (2 x) - \int \frac{x}{2} d x]_{\frac{1}{2}}^{7}

\int \frac{x}{2} d x = \frac{x ^{2}}{4}

= 16 [\frac{1}{2} x^{2} ln (2 x) - \frac{x ^{2}}{4}]_{\frac{1}{2}}^{7}

Compute the boundaries:

- \frac{195}{16} + \frac{49}{2} ln (14)

= 16 (- \frac{195}{16} + \frac{49}{2} ln (14))

Simplify

= - 195 + 392 ln (14)

f (x) = 2 + x

x \to 2 lim (\frac{2 x - 3}{x - 2})

\frac{\partial}{\partial x} (e^{m x})

x sin (2 x)

\frac{8}{x}

What is the integral from 1/2 to 7 of 16xln(2x) ?
The integral from 1/2 to 7 of 16xln(2x) is -195+392ln(14)