What is the integral of (ln(x))^6 ?

The integral of (ln(x))^6 is xln^6(x)-6(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))^6

\int (ln (x))^{6} d x

x ln^{6} (x) - 6 (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))^{6} d x

Simplify

= \int ln^{6} (x) d x

Apply u-substitution

: \int u^{6} e^{u} d u

= \int u^{6} e^{u} d u

Apply Integration By Parts

: u^{6} e^{u} - \int 6 u^{5} e^{u} d u

= u^{6} e^{u} - \int 6 u^{5} e^{u} d u

\int 6 u^{5} e^{u} d u = 6 (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))))

= u^{6} e^{u} - 6 (u^{5} e^{u} - 5 (u^{4} e^{u} - 4 (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))))

Substitute back

u = ln (x)

= ln^{6} (x) e^{l n (x)} - 6 (ln^{5} (x) e^{l n (x)} - 5 (ln^{4} (x) e^{l n (x)} - 4 (ln^{3} (x) e^{l n (x)} - 3 (ln^{2} (x) e^{l n (x)} - 2 (e^{l n (x)} ln (x) - e^{l n (x)})))))

Simplify

ln^{6} (x) e^{l n (x)} - 6 (ln^{5} (x) e^{l n (x)} - 5 (ln^{4} (x) e^{l n (x)} - 4 (ln^{3} (x) e^{l n (x)} - 3 (ln^{2} (x) e^{l n (x)} - 2 (e^{l n (x)} ln (x) - e^{l n (x)}))))) : x ln^{6} (x) - 6 (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))))

= x ln^{6} (x) - 6 (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^{6} (x) - 6 (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

\frac{d}{d x} ((x^{2} + 3) 4 - x^{2})

(cos^{3} (t))^{^{'}}

\frac{d}{d x} (sin (1000 x))

\int \frac{3}{2 x ^{2} + x - 10} d x

\int x (x^{2} + 1)^{6} d x

What is the integral of (ln(x))^6 ?
The integral of (ln(x))^6 is xln^6(x)-6(xln^5(x)-5(xln^4(x)-4(xln^3(x)-3(xln^2(x)-2(xln(x)-x)))))+C