What is the integral of 8x(ln(x^5))^2 ?

The integral of 8x(ln(x^5))^2 is 8(25/2 x^2ln^2(x)-5(5/2 x^2ln(x)-(5x^2)/4))+C

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integral of 8x(ln(x^5))^2

\int 8 x (ln (x^{5}))^{2} d x

8 (\frac{25}{2} x^{2} ln^{2} (x) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4})) + C

Solution steps

\int 8 x (ln (x^{5}))^{2} d x

Take the constant out:

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

= 8 \cdot \int x (ln (x^{5}))^{2} d x

Simplify

= 8 \cdot \int x ln^{2} (x^{5}) d x

Apply Integration By Parts

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

= 8 (\frac{1}{2} x^{2} ln^{2} (x^{5}) - \int 5 x ln (x^{5}) d x)

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

= 8 (\frac{1}{2} x^{2} ln^{2} (x^{5}) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4}))

\frac{1}{2} x^{2} ln^{2} (x^{5}) = \frac{25}{2} x^{2} ln^{2} (x)

= 8 (\frac{25}{2} x^{2} ln^{2} (x) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4}))

Add a constant to the solution

= 8 (\frac{25}{2} x^{2} ln^{2} (x) - 5 (\frac{5}{2} x^{2} ln (x) - \frac{5 x ^{2}}{4})) + C

\frac{d y}{d x}, x y = 0

\int_{0}^{π} cos^{2} (x) d x

\frac{d x}{d t} = x - 2 x^{2}

\frac{1}{5}

x^{2}, x, [\frac{1}{4}, 1]

What is the integral of 8x(ln(x^5))^2 ?
The integral of 8x(ln(x^5))^2 is 8(25/2 x^2ln^2(x)-5(5/2 x^2ln(x)-(5x^2)/4))+C