What is the integral from 1 to 2 of 2xln(3x) ?

The integral from 1 to 2 of 2xln(3x) is ln(432)-3/2

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integral from 1 to 2 of 2xln(3x)

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

ln (432) - \frac{3}{2}

Solution steps

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

Take the constant out:

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

= 2 \cdot \int_{1}^{2} x ln (3 x) d x

Apply Integration By Parts

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

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

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

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

Compute the boundaries:

2 ln (6) - \frac{3}{4} - \frac{1}{2} ln (3)

= 2 (2 ln (6) - \frac{3}{4} - \frac{1}{2} ln (3))

Simplify

= ln (432) - \frac{3}{2}

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What is the integral from 1 to 2 of 2xln(3x) ?
The integral from 1 to 2 of 2xln(3x) is ln(432)-3/2