What is the integral of 3/(xln(2x)) ?

The integral of 3/(xln(2x)) is 3ln|ln(2x)|+C

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integral of 3/(xln(2x))

\int \frac{3}{x ln ( 2 x )} d x

3 ln ∣ ln (2 x) ∣ + C

Solution steps

\int \frac{3}{x ln ( 2 x )} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{x ln ( 2 x )} d x

Apply u-substitution

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

= 3 \cdot \int \frac{1}{u ln ( u )} d u

Apply u-substitution

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

= 3 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 ln ∣ v ∣

Substitute back

= 3 ln ∣ ln (2 x) ∣

Add a constant to the solution

= 3 ln ∣ ln (2 x) ∣ + C

\int 1 d x^{2}

y = 2 x + 6

\int \frac{20 x ^{3} - 40 x ^{2} + 7}{x ^{2} - 2 x} d x

f (x) = 3 e^{- x^{2}}

\int cos (ln (5 x)) d x

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