What is the integral of ((1+2ln(x))^4)/x ?

The integral of ((1+2ln(x))^4)/x is 1/10 (1+2ln(x))^5+C

integral of ((1+2ln(x))^4)/x

\int \frac{( 1 + 2 ln ( x ) ) ^{4}}{x} d x

\frac{1}{10} (1 + 2 ln (x))^{5} + C

Solution steps

\int \frac{( 1 + 2 ln ( x ) ) ^{4}}{x} d x

Apply u-substitution

= \int \frac{u ^{4}}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u^{4} d u

Apply the Power Rule

= \frac{1}{2} \cdot \frac{u ^{5}}{5}

Substitute back

u = 1 + 2 ln (x)

= \frac{1}{2} \cdot \frac{( 1 + 2 ln ( x ) ) ^{5}}{5}

Simplify

\frac{1}{2} \cdot \frac{( 1 + 2 ln ( x ) ) ^{5}}{5} : \frac{1}{10} (1 + 2 ln (x))^{5}

= \frac{1}{10} (1 + 2 ln (x))^{5}

Add a constant to the solution

= \frac{1}{10} (1 + 2 ln (x))^{5} + C

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What is the integral of ((1+2ln(x))^4)/x ?
The integral of ((1+2ln(x))^4)/x is 1/10 (1+2ln(x))^5+C