What is the integral of 5/(xln^3(x)) ?

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

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integral of 5/(xln^3(x))

\int \frac{5}{x ln ^{3} ( x )} d x

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

Solution steps

\int \frac{5}{x ln ^{3} ( 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{1}{x ln ^{3} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{3}} d u

= 5 \cdot \int \frac{1}{u ^{3}} d u

Apply the Power Rule

: - \frac{1}{2 u ^{2}}

= 5 (- \frac{1}{2 u ^{2}})

Substitute back

u = ln (x)

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

Simplify

5 (- \frac{1}{2 ln ^{2} ( x )}) : - \frac{5}{2 ln ^{2} ( x )}

= - \frac{5}{2 ln ^{2} ( x )}

Add a constant to the solution

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

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