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

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

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integral of 5xln(x^3)

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

5 (\frac{3}{2} x^{2} ln (x) - \frac{3 x ^{2}}{4}) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

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

Add a constant to the solution

= 5 (\frac{3}{2} x^{2} ln (x) - \frac{3 x ^{2}}{4}) + C

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