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

The integral of ln^6(x) 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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of ln^6(x)

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

\int sin (3 x - 5) d x

\frac{d}{d θ} (\frac{5}{sin ( θ ) - 2 cos ( θ )})

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

y^{^{'}} = 9 x + y, y (0) = 8

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