What is the integral of 2x^3ln(5x) ?

The integral of 2x^3ln(5x) is 2(1/4 x^4ln(5x)-(x^4)/(16))+C

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integral of 2x^3ln(5x)

\int 2 x^{3} ln (5 x) d x

2 (\frac{1}{4} x^{4} ln (5 x) - \frac{x ^{4}}{16}) + C

Solution steps

\int 2 x^{3} ln (5 x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x^{3} ln (5 x) d x

Apply Integration By Parts

: \frac{1}{4} x^{4} ln (5 x) - \int \frac{x ^{3}}{4} d x

= 2 (\frac{1}{4} x^{4} ln (5 x) - \int \frac{x ^{3}}{4} d x)

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

= 2 (\frac{1}{4} x^{4} ln (5 x) - \frac{x ^{4}}{16})

Add a constant to the solution

= 2 (\frac{1}{4} x^{4} ln (5 x) - \frac{x ^{4}}{16}) + C

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What is the integral of 2x^3ln(5x) ?
The integral of 2x^3ln(5x) is 2(1/4 x^4ln(5x)-(x^4)/(16))+C