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

The integral of ((x^2)/3+1)ln(2x) is 1/3 (1/3 x^3ln(2x)-(x^3)/9)+xln(2x)-x+C

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

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

\frac{1}{3} (\frac{1}{3} x^{3} ln (2 x) - \frac{x ^{3}}{9}) + x ln (2 x) - x + C

Solution steps

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

Expand

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

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

\int \frac{x ^{2} ln ( 2 x )}{3} d x = \frac{1}{3} (\frac{1}{3} x^{3} ln (2 x) - \frac{x ^{3}}{9})

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

= \frac{1}{3} (\frac{1}{3} x^{3} ln (2 x) - \frac{x ^{3}}{9}) + x ln (2 x) - x

Add a constant to the solution

= \frac{1}{3} (\frac{1}{3} x^{3} ln (2 x) - \frac{x ^{3}}{9}) + x ln (2 x) - x + C

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What is the integral of ((x^2)/3+1)ln(2x) ?
The integral of ((x^2)/3+1)ln(2x) is 1/3 (1/3 x^3ln(2x)-(x^3)/9)+xln(2x)-x+C