What is the integral of 4/(xln(2x^2)) ?

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

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integral of 4/(xln(2x^2))

\int \frac{4}{x ln ( 2 x ^{2} )} d x

2 ln ∣ ln (2) + 2 ln (x) ∣ + C

Solution steps

\int \frac{4}{x ln ( 2 x ^{2} )} 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 ( 2 x ^{2} )} d x

Apply u-substitution

: \int \frac{1}{2 u ln ( 2 u )} d u

= 4 \cdot \int \frac{1}{2 u ln ( 2 u )} d u

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{u ln ( 2 u )} d u

Apply u-substitution

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

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{v ln ( v )} d v

Apply u-substitution

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

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{w} d w

Use the common integral:

\int \frac{1}{w} d w = ln (∣ w ∣)

= 4 \cdot \frac{1}{2} ln ∣ w ∣

Substitute back

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

Simplify

4 \cdot \frac{1}{2} ln ln (2 x^{2}) : 2 ln ∣ ln (2) + 2 ln (x) ∣

= 2 ln ∣ ln (2) + 2 ln (x) ∣

Add a constant to the solution

= 2 ln ∣ ln (2) + 2 ln (x) ∣ + C

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