What is the integral of 8/(xln(2x)) ?

The integral of 8/(xln(2x)) is 8ln|ln(2x)|+C

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integral of 8/(xln(2x))

\int \frac{8}{x ln ( 2 x )} d x

8 ln ∣ ln (2 x) ∣ + C

Solution steps

\int \frac{8}{x ln ( 2 x )} d x

Take the constant out:

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

= 8 \cdot \int \frac{1}{x ln ( 2 x )} d x

Apply u-substitution

: \int \frac{1}{u ln ( u )} d u

= 8 \cdot \int \frac{1}{u ln ( u )} d u

Apply u-substitution

: \int \frac{1}{v} d v

= 8 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 8 ln ∣ v ∣

Substitute back

= 8 ln ∣ ln (2 x) ∣

Add a constant to the solution

= 8 ln ∣ ln (2 x) ∣ + C

\frac{\partial}{\partial x} (e^{- x y})

x \to + 2 + lim (x^{2})

\frac{d}{d x} (z arctan (\frac{y}{x}))

\frac{e ^{x}}{e ^{x} + 1}

y^{^{''}} + 5 y = - 180 x^{2} e^{5 x}

What is the integral of 8/(xln(2x)) ?
The integral of 8/(xln(2x)) is 8ln|ln(2x)|+C