What is the integral of (ln(x))^3 ?

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

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integral of (ln(x))^3

integral

(ln (x))^{3}

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

Solution steps

\int (ln (x))^{3} d x

Apply Integration By Parts

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

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

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

= x ln^{3} (x) - 3 (x ln^{2} (x) - 2 (x ln (x) - x))

Add a constant to the solution

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

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

\frac{ln ( t )}{( t + 1 ) ^{2}}

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

(u^{2}) (1 + u^{2})^{0.5}

6^{x}

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