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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

\int \frac{9}{( t ^{2} + 1 ) ^{\frac{3}{2}}} d t

y = x^{5 x}

\int_{- 1}^{1} (3 t - 5) d t

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

\int (3 x + 1) (4 - x) d x

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