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

The integral of x^3ln(5x^2) is 1/4 x^4(ln(5)+2ln(x))-(x^4)/8+C

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integral of x^3ln(5x^2)

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

\frac{1}{4} x^{4} (ln (5) + 2 ln (x)) - \frac{x ^{4}}{8} + C

Solution steps

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

Apply Integration By Parts

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

\int \frac{x ^{3}}{2} d x = \frac{x ^{4}}{8}

= \frac{1}{4} x^{4} ln (5 x^{2}) - \frac{x ^{4}}{8}

Simplify

ln (5 x^{2}) : 2 ln (x) + ln (5)

= \frac{1}{4} x^{4} (ln (5) + 2 ln (x)) - \frac{x ^{4}}{8}

Add a constant to the solution

= \frac{1}{4} x^{4} (ln (5) + 2 ln (x)) - \frac{x ^{4}}{8} + C

What is the integral of x^3ln(5x^2) ?
The integral of x^3ln(5x^2) is 1/4 x^4(ln(5)+2ln(x))-(x^4)/8+C