What is the integral of 4/(xln^3(x)) ?

The integral of 4/(xln^3(x)) is -2/(ln^2(x))+C

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integral of 4/(xln^3(x))

\int \frac{4}{x ln ^{3} ( x )} d x

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

Solution steps

\int \frac{4}{x ln ^{3} ( 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{1}{x ln ^{3} ( x )} d x

Apply u-substitution

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

= 4 \cdot \int \frac{1}{u ^{3}} d u

Apply the Power Rule

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

= 4 (- \frac{1}{2 u ^{2}})

Substitute back

u = ln (x)

= 4 (- \frac{1}{2 ln ^{2} ( x )})

Simplify

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

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

Add a constant to the solution

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

e^{- (t - 3)}

x \to 0 + lim (\frac{4}{5 x ^{\frac{1}{3}}})

\frac{d y}{d x} = \frac{y + x ^{2} + y ^{2}}{x}

\frac{d}{d x} ((2 x^{3} - 5) (x^{2} - 5 ln (x)))

\frac{d}{d x} (\frac{x}{csc ( x )})

What is the integral of 4/(xln^3(x)) ?
The integral of 4/(xln^3(x)) is -2/(ln^2(x))+C