What is the integral from 1/5 to 3 of 12xln(5x) ?

The integral from 1/5 to 3 of 12xln(5x) is 12(-56/25+9/2 ln(15))

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integral from 1/5 to 3 of 12xln(5x)

\int_{\frac{1}{5}}^{3} 12 x ln (5 x) d x

12 (- \frac{56}{25} + \frac{9}{2} ln (15))

Decimal Notation

119.35471 \dots

Solution steps

\int_{\frac{1}{5}}^{3} 12 x ln (5 x) d x

Take the constant out:

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

= 12 \cdot \int_{\frac{1}{5}}^{3} x ln (5 x) d x

Apply Integration By Parts

: [\frac{1}{2} x^{2} ln (5 x) - \int \frac{x}{2} d x]_{\frac{1}{5}}^{3}

= 12 [\frac{1}{2} x^{2} ln (5 x) - \int \frac{x}{2} d x]_{\frac{1}{5}}^{3}

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

= 12 [\frac{1}{2} x^{2} ln (5 x) - \frac{x ^{2}}{4}]_{\frac{1}{5}}^{3}

Compute the boundaries:

- \frac{56}{25} + \frac{9}{2} ln (15)

= 12 (- \frac{56}{25} + \frac{9}{2} ln (15))

\int (x^{2} + x + \frac{1}{x}) d x

f (x) = 5 x^{2} - 5 x + 9

\frac{d}{d x} (- 9 x^{3} + 27 x + 7 x - 66)

3 x, at x = 4

\int 5 tan^{3} (x) sec^{3} (x) d x

What is the integral from 1/5 to 3 of 12xln(5x) ?
The integral from 1/5 to 3 of 12xln(5x) is 12(-56/25+9/2 ln(15))