What is the integral of 8/(xln(5x)) ?

The integral of 8/(xln(5x)) is 8ln|ln(5x)|+C

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integral of 8/(xln(5x))

\int \frac{8}{x ln ( 5 x )} d x

8 ln ∣ ln (5 x) ∣ + C

Solution steps

\int \frac{8}{x ln ( 5 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 ( 5 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 (5 x) ∣

Add a constant to the solution

= 8 ln ∣ ln (5 x) ∣ + C

n = 1 \sum \infty \frac{1}{3 n - 1}

\int \frac{1}{x ^{2} x ^{2} - 225} d x

\frac{d}{d x} (cot (9 x))

f (x) = - 9 cos (3 x)

\frac{d y}{d x}, ln (x^{2} + y^{2}) = 2 (arctan (\frac{y}{x}))

What is the integral of 8/(xln(5x)) ?
The integral of 8/(xln(5x)) is 8ln|ln(5x)|+C