What is the integral of (3x^2+2)ln(2x) ?

The integral of (3x^2+2)ln(2x) is x^3ln(2x)+2xln(2x)-(x^3)/3-2x+C

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integral of (3x^2+2)ln(2x)

\int (3 x^{2} + 2) ln (2 x) d x

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

Solution steps

\int (3 x^{2} + 2) ln (2 x) d x

Apply Integration By Parts

= ln (2 x) (x^{3} + 2 x) - \int x^{2} + 2 d x

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

= ln (2 x) (x^{3} + 2 x) - (\frac{x ^{3}}{3} + 2 x)

Simplify

ln (2 x) (x^{3} + 2 x) - (\frac{x ^{3}}{3} + 2 x) : x^{3} ln (2 x) + 2 x ln (2 x) - \frac{x ^{3}}{3} - 2 x

= x^{3} ln (2 x) + 2 x ln (2 x) - \frac{x ^{3}}{3} - 2 x

Add a constant to the solution

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

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