What is the integral of 4x(ln(x^9))^2 ?

The integral of 4x(ln(x^9))^2 is 324(1/2 x^2ln^2(x)-1/2 x^2ln(x)+(x^2)/4)+C

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integral of 4x(ln(x^9))^2

\int 4 x (ln (x^{9}))^{2} d x

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

Solution steps

\int 4 x (ln (x^{9}))^{2} d x

Take the constant out:

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

= 4 \cdot \int x (ln (x^{9}))^{2} d x

Simplify

x (ln (x^{9}))^{2} : 81 x ln^{2} (x)

= 4 \cdot \int 81 x ln^{2} (x) d x

Take the constant out:

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

= 4 \cdot 81 \cdot \int x ln^{2} (x) d x

Apply Integration By Parts

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

= 4 \cdot 81 (\frac{1}{2} x^{2} ln^{2} (x) - \int x ln (x) d x)

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

= 4 \cdot 81 (\frac{1}{2} x^{2} ln^{2} (x) - (\frac{1}{2} x^{2} ln (x) - \frac{x ^{2}}{4}))

Simplify

= 324 (\frac{1}{2} x^{2} ln^{2} (x) - \frac{1}{2} x^{2} ln (x) + \frac{x ^{2}}{4})

Add a constant to the solution

= 324 (\frac{1}{2} x^{2} ln^{2} (x) - \frac{1}{2} x^{2} ln (x) + \frac{x ^{2}}{4}) + C

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What is the integral of 4x(ln(x^9))^2 ?
The integral of 4x(ln(x^9))^2 is 324(1/2 x^2ln^2(x)-1/2 x^2ln(x)+(x^2)/4)+C