What is the integral of 4/(xln(5x)) ?

The integral of 4/(xln(5x)) is 4ln|ln(5x)|+C

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integral of 4/(xln(5x))

\int \frac{4}{x ln ( 5 x )} d x

4 ln ∣ ln (5 x) ∣ + C

Solution steps

\int \frac{4}{x ln ( 5 x )} 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 ( 5 x )} d x

Apply u-substitution

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

Apply u-substitution

= 4 \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 4 ln ∣ v ∣

Substitute back

= 4 ln ∣ ln (5 x) ∣

Add a constant to the solution

= 4 ln ∣ ln (5 x) ∣ + C

\int \frac{1}{( 441 + x ^{2} ) ^{\frac{3}{2}}} d x

x^{^{'}} = k (300 - x)

\frac{d}{d x} (\frac{8 x ^{3} - 9}{x ^{2}})

\int \frac{1}{u ^{5}} d u

x^{2} y^{^{''}} - 3 x y^{^{'}} + 3 y = 0, y (1) = 5, y^{^{'}} (1) = 3

What is the integral of 4/(xln(5x)) ?
The integral of 4/(xln(5x)) is 4ln|ln(5x)|+C