What is the integral of 9x(ln(x^3))^2 ?

The integral of 9x(ln(x^3))^2 is 9(9/2 x^2ln^2(x)-3(3/2 x^2ln(x)-(3x^2)/4))+C

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integral of 9x(ln(x^3))^2

\int 9 x (ln (x^{3}))^{2} d x

9 (\frac{9}{2} x^{2} ln^{2} (x) - 3 (\frac{3}{2} x^{2} ln (x) - \frac{3 x ^{2}}{4})) + C

Solution steps

\int 9 x (ln (x^{3}))^{2} d x

Take the constant out:

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

= 9 \cdot \int x (ln (x^{3}))^{2} d x

Simplify

= 9 \cdot \int x ln^{2} (x^{3}) d x

Apply Integration By Parts

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

= 9 (\frac{1}{2} x^{2} ln^{2} (x^{3}) - \int 3 x ln (x^{3}) d x)

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

= 9 (\frac{1}{2} x^{2} ln^{2} (x^{3}) - 3 (\frac{3}{2} x^{2} ln (x) - \frac{3 x ^{2}}{4}))

\frac{1}{2} x^{2} ln^{2} (x^{3}) = \frac{9}{2} x^{2} ln^{2} (x)

= 9 (\frac{9}{2} x^{2} ln^{2} (x) - 3 (\frac{3}{2} x^{2} ln (x) - \frac{3 x ^{2}}{4}))

Add a constant to the solution

= 9 (\frac{9}{2} x^{2} ln^{2} (x) - 3 (\frac{3}{2} x^{2} ln (x) - \frac{3 x ^{2}}{4})) + C

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What is the integral of 9x(ln(x^3))^2 ?
The integral of 9x(ln(x^3))^2 is 9(9/2 x^2ln^2(x)-3(3/2 x^2ln(x)-(3x^2)/4))+C