zs

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

受欢迎的微积分 >

integral of (ln(x))^6

解答

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

解答

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

求解步骤

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

化简

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

使用换元积分法

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

使用分布积分法

= 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})))))

u = ln (x)

代回

= 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)})))))

化简

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)))))

解答补常数

= 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

作图

流行的例子

derivative of (x^2+3sqrt(4-x^2))

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

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

derivative of sin(1000x)

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

integral of 3/(2x^2+x-10)

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

integral of x(x^2+1)^6

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