What is the integral of 3x(ln(x^7))^2 ?

The integral of 3x(ln(x^7))^2 is 3(49/2 x^2ln^2(x)-7(7/2 x^2ln(x)-(7x^2)/4))+C

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integral of 3x(ln(x^7))^2

\int 3 x (ln (x^{7}))^{2} d x

3 (\frac{49}{2} x^{2} ln^{2} (x) - 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4})) + C

Solution steps

\int 3 x (ln (x^{7}))^{2} d x

Take the constant out:

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

= 3 \cdot \int x (ln (x^{7}))^{2} d x

Apply Integration By Parts

= 3 (\frac{1}{2} x^{2} ln^{2} (x^{7}) - \int 7 x ln (x^{7}) d x)

\int 7 x ln (x^{7}) d x = 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4})

= 3 (\frac{1}{2} x^{2} ln^{2} (x^{7}) - 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4}))

\frac{1}{2} x^{2} ln^{2} (x^{7}) = \frac{49}{2} x^{2} ln^{2} (x)

= 3 (\frac{49}{2} x^{2} ln^{2} (x) - 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4}))

Add a constant to the solution

= 3 (\frac{49}{2} x^{2} ln^{2} (x) - 7 (\frac{7}{2} x^{2} ln (x) - \frac{7 x ^{2}}{4})) + C

What is the integral of 3x(ln(x^7))^2 ?
The integral of 3x(ln(x^7))^2 is 3(49/2 x^2ln^2(x)-7(7/2 x^2ln(x)-(7x^2)/4))+C