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

The integral of x^3*ln(2x) is 1/4 x^4ln(2x)-(x^4)/(16)+C

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integral of x^3*ln(2x)

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

\frac{1}{4} x^{4} ln (2 x) - \frac{x ^{4}}{16} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x ^{3}}{4} d x = \frac{x ^{4}}{16}

= \frac{1}{4} x^{4} ln (2 x) - \frac{x ^{4}}{16}

Add a constant to the solution

= \frac{1}{4} x^{4} ln (2 x) - \frac{x ^{4}}{16} + C

\frac{s + 5}{s ( s ^{2} + 5 s + 6 )}

y^{^{'''}} - y^{^{''}} = x e^{x}

(x^{2} + 4) y = 8, (- 2, 1)

\frac{( s + 2 )}{s ^{2} + s + 1}

\frac{d}{d x} (\frac{d y}{d x}) + 9 y = 0

What is the integral of x^3*ln(2x) ?
The integral of x^3*ln(2x) is 1/4 x^4ln(2x)-(x^4)/(16)+C