What is the integral of 5/(xln(x^2)) ?

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

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integral of 5/(xln(x^2))

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

\frac{5}{2} ln (2 ∣ ln (x) ∣) + C

Solution steps

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

Apply u-substitution

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

= 5 \cdot \int \frac{1}{2 u ln ( u )} d u

Take the constant out:

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

= 5 \cdot \frac{1}{2} \cdot \int \frac{1}{u ln ( u )} d u

Apply u-substitution

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

= 5 \cdot \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 5 \cdot \frac{1}{2} ln ∣ v ∣

Substitute back

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

Simplify

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

= \frac{5}{2} ln (2 ∣ ln (x) ∣)

Add a constant to the solution

= \frac{5}{2} ln (2 ∣ ln (x) ∣) + C

\int \frac{6 x ^{3} - 9 x + 7}{x ^{3} - x ^{2}} d x

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\int \frac{x + 7}{x} d x

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