What is the integral of 6xln(x^6) ?

The integral of 6xln(x^6) is 6(3x^2ln(x)-(3x^2)/2)+C

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integral of 6xln(x^6)

\int 6 x ln (x^{6}) d x

6 (3 x^{2} ln (x) - \frac{3 x ^{2}}{2}) + C

Solution steps

\int 6 x ln (x^{6}) d x

Take the constant out:

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

= 6 \cdot \int x ln (x^{6}) d x

Apply Integration By Parts

= 6 (\frac{1}{2} x^{2} ln (x^{6}) - \int 3 x d x)

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

= 6 (\frac{1}{2} x^{2} ln (x^{6}) - \frac{3 x ^{2}}{2})

\frac{1}{2} x^{2} ln (x^{6}) = 3 x^{2} ln (x)

= 6 (3 x^{2} ln (x) - \frac{3 x ^{2}}{2})

Add a constant to the solution

= 6 (3 x^{2} ln (x) - \frac{3 x ^{2}}{2}) + C

What is the integral of 6xln(x^6) ?
The integral of 6xln(x^6) is 6(3x^2ln(x)-(3x^2)/2)+C