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

The integral from 1 to 2 of xln((2x)^2) is 7ln(2)-3/2

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

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

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

Decimal Notation

3.35203 \dots

Solution steps

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

Simplify

x ln ((2 x)^{2}) : x ln (4 x^{2})

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

Apply Integration By Parts

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

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

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

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

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

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

Compute the boundaries:

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

= 7 ln (2) - \frac{3}{2}

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What is the integral from 1 to 2 of xln((2x)^2) ?
The integral from 1 to 2 of xln((2x)^2) is 7ln(2)-3/2