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

The integral of ((ln(x))^5)/5 is 1/5 xln^5(x)-xln^4(x)+4xln^3(x)-12xln^2(x)+24xln(x)-24x+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of ((ln(x))^5)/5

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

\frac{1}{5} x ln^{5} (x) - x ln^{4} (x) + 4 x ln^{3} (x) - 12 x ln^{2} (x) + 24 x ln (x) - 24 x + C

Solution steps

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

Take the constant out:

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

= \frac{1}{5} \cdot \int (ln (x))^{5} d x

Simplify

= \frac{1}{5} \cdot \int ln^{5} (x) d x

Apply Integration By Parts

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

= \frac{1}{5} (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))))

= \frac{1}{5} (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x)))))

Simplify

\frac{1}{5} (x ln^{5} (x) - 5 (x ln^{4} (x) - 4 (x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))))) : \frac{1}{5} x ln^{5} (x) - x ln^{4} (x) + 4 x ln^{3} (x) - 12 x ln^{2} (x) + 24 x ln (x) - 24 x

= \frac{1}{5} x ln^{5} (x) - x ln^{4} (x) + 4 x ln^{3} (x) - 12 x ln^{2} (x) + 24 x ln (x) - 24 x

Add a constant to the solution

= \frac{1}{5} x ln^{5} (x) - x ln^{4} (x) + 4 x ln^{3} (x) - 12 x ln^{2} (x) + 24 x ln (x) - 24 x + C

\int \frac{x ^{2} - 1}{x + 1} d x

\int (1 - x)^{30} d x

\int sec^{2} (3 t) d t

\int 9 (3 x - 1)^{2} cos (3 x) d x

\int \frac{x}{( x ^{2} + 9 ) ^{\frac{3}{2}}} d x

What is the integral of ((ln(x))^5)/5 ?
The integral of ((ln(x))^5)/5 is 1/5 xln^5(x)-xln^4(x)+4xln^3(x)-12xln^2(x)+24xln(x)-24x+C