What is the integral of ((1+6ln(x))^2)/x ?

The integral of ((1+6ln(x))^2)/x is 1/18 (1+6ln(x))^3+C

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integral of ((1+6ln(x))^2)/x

\int \frac{( 1 + 6 ln ( x ) ) ^{2}}{x} d x

\frac{1}{18} (1 + 6 ln (x))^{3} + C

Solution steps

\int \frac{( 1 + 6 ln ( x ) ) ^{2}}{x} d x

Apply u-substitution

= \int \frac{u ^{2}}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int u^{2} d u

Apply the Power Rule

= \frac{1}{6} \cdot \frac{u ^{3}}{3}

Substitute back

u = 1 + 6 ln (x)

= \frac{1}{6} \cdot \frac{( 1 + 6 ln ( x ) ) ^{3}}{3}

Simplify

\frac{1}{6} \cdot \frac{( 1 + 6 ln ( x ) ) ^{3}}{3} : \frac{1}{18} (1 + 6 ln (x))^{3}

= \frac{1}{18} (1 + 6 ln (x))^{3}

Add a constant to the solution

= \frac{1}{18} (1 + 6 ln (x))^{3} + C

What is the integral of ((1+6ln(x))^2)/x ?
The integral of ((1+6ln(x))^2)/x is 1/18 (1+6ln(x))^3+C