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

The integral of (ln(x))^5 is xln^5(x)-5(xln^4(x)-4(xln^3(x)-3(xln^2(x)-2(xln(x)-x))))+C

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integral of (ln(x))^5

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

x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))) + C

Solution steps

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

Simplify

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

Apply Integration By Parts

: x ln^{5} (x) - \int 5 ln^{4} (x) d x

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

\int 5 ln^{4} (x) d x = 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))

= x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))

Add a constant to the solution

= x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))) + C

t^{3} \frac{d y}{d t} + 3 t^{2} y = 8 cos (t), y (π) = 0

y = (x^{2} - 2 x + 2) e^{\frac{5 x}{2}}

\int sec (θ) tan (θ) d θ

\frac{d}{d t} (p (t) e^{t})

y = 5 + 1 - x, (- 8, 8)

What is the integral of (ln(x))^5 ?
The integral of (ln(x))^5 is xln^5(x)-5(xln^4(x)-4(xln^3(x)-3(xln^2(x)-2(xln(x)-x))))+C