What is the integral of 5((ln(x))^2)/x ?

The integral of 5((ln(x))^2)/x is 5/3 ln^3(x)+C

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integral of 5((ln(x))^2)/x

\int 5 \frac{( ln ( x ) ) ^{2}}{x} d x

\frac{5}{3} ln^{3} (x) + C

Solution steps

\int 5 \cdot \frac{( ln ( x ) ) ^{2}}{x} d x

Take the constant out:

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

= 5 \cdot \int \frac{( ln ( x ) ) ^{2}}{x} d x

Apply u-substitution

: \int u^{2} d u

= 5 \cdot \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= 5 \cdot \frac{u ^{3}}{3}

Substitute back

u = ln (x)

= 5 \cdot \frac{ln ^{3} ( x )}{3}

Simplify

5 \cdot \frac{ln ^{3} ( x )}{3} : \frac{5}{3} ln^{3} (x)

= \frac{5}{3} ln^{3} (x)

Add a constant to the solution

= \frac{5}{3} ln^{3} (x) + C

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What is the integral of 5((ln(x))^2)/x ?
The integral of 5((ln(x))^2)/x is 5/3 ln^3(x)+C