What is the integral from 2 to 4 of 3x^2ln(2x) ?

The integral from 2 to 4 of 3x^2ln(2x) is -56/3+176ln(2)

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integral from 2 to 4 of 3x^2ln(2x)

\int_{2}^{4} 3 x^{2} ln (2 x) d x

- \frac{56}{3} + 176 ln (2)

Solution steps

\int_{2}^{4} 3 x^{2} ln (2 x) d x

Take the constant out:

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

= 3 \cdot \int_{2}^{4} x^{2} ln (2 x) d x

Apply Integration By Parts

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

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

\int \frac{x ^{2}}{3} d x = \frac{x ^{3}}{9}

= 3 [\frac{1}{3} x^{3} ln (2 x) - \frac{x ^{3}}{9}]_{2}^{4}

Compute the boundaries:

- \frac{56}{9} + \frac{176}{3} ln (2)

= 3 (- \frac{56}{9} + \frac{176}{3} ln (2))

Simplify

= - \frac{56}{3} + 176 ln (2)

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What is the integral from 2 to 4 of 3x^2ln(2x) ?
The integral from 2 to 4 of 3x^2ln(2x) is -56/3+176ln(2)