What is the integral of (ln(x))^4 ?

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

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integral of (ln(x))^4

integral

(ln (x))^{4}

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))^{4} d x

Simplify

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

Apply Integration By Parts

: x ln^{4} (x) - \int 4 ln^{3} (x) d x

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

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

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

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

\frac{u}{( 1 + u ^{2} )}

\frac{2 tan ( x )}{( 1 + 2 tan ^{2} ( x ) )}

\frac{( x + 1 ) ^{2}}{( x - 1 ) ^{2}}

tan (2 - 3 x)

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