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

The integral of (-4(ln(x))^4)/x is -4/5 ln^5(x)+C

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integral of (-4(ln(x))^4)/x

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

- \frac{4}{5} ln^{5} (x) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= - 4 \cdot \int u^{4} d u

Apply the Power Rule

= - 4 \cdot \frac{u ^{5}}{5}

Substitute back

u = ln (x)

= - 4 \cdot \frac{ln ^{5} ( x )}{5}

Simplify

- 4 \cdot \frac{ln ^{5} ( x )}{5} : - \frac{4}{5} ln^{5} (x)

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

Add a constant to the solution

= - \frac{4}{5} ln^{5} (x) + C

What is the integral of (-4(ln(x))^4)/x ?
The integral of (-4(ln(x))^4)/x is -4/5 ln^5(x)+C