What is the integral of (1+3x^2)ln(2x) ?

The integral of (1+3x^2)ln(2x) is xln(2x)-x+x^3ln(2x)-(x^3)/3+C

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integral of (1+3x^2)ln(2x)

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

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

Solution steps

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

Expand

(1 + 3 x^{2}) ln (2 x) : ln (2 x) + 3 x^{2} ln (2 x)

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int ln (2 x) d x = x ln (2 x) - x

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

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

Add a constant to the solution

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

f (x) = 12 x - 2 x

x \frac{d y}{d x} = 2 y

t^{2} cos (- 2 t)

\int_{0}^{1} \frac{2}{1 + x ^{2}} d x

y^{^{'}} = \frac{1}{e ^{y} - x}

What is the integral of (1+3x^2)ln(2x) ?
The integral of (1+3x^2)ln(2x) is xln(2x)-x+x^3ln(2x)-(x^3)/3+C