What is the integral of 4xln(8x^3) ?

The integral of 4xln(8x^3) is 6x^2ln(2x)-3x^2+C

integral of 4xln(8x^3)

\int 4 x ln (8 x^{3}) d x

6 x^{2} ln (2 x) - 3 x^{2} + C

Solution steps

\int 4 x ln (8 x^{3}) d x

Take the constant out:

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

= 4 \cdot \int x ln (8 x^{3}) d x

Apply Integration By Parts

= 4 (\frac{1}{2} x^{2} ln (8 x^{3}) - \int \frac{3 x}{2} d x)

\int \frac{3 x}{2} d x = \frac{3 x ^{2}}{4}

= 4 (\frac{1}{2} x^{2} ln (8 x^{3}) - \frac{3 x ^{2}}{4})

Simplify

4 (\frac{1}{2} x^{2} ln (8 x^{3}) - \frac{3 x ^{2}}{4}) : 6 x^{2} ln (2 x) - 3 x^{2}

= 6 x^{2} ln (2 x) - 3 x^{2}

Add a constant to the solution

= 6 x^{2} ln (2 x) - 3 x^{2} + C

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What is the integral of 4xln(8x^3) ?
The integral of 4xln(8x^3) is 6x^2ln(2x)-3x^2+C