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

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

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integral of 2/(xln(x))

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

2 ln ∣ ln (x) ∣ + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

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

= 2 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = ln (x)

= 2 ln ∣ ln (x) ∣

Add a constant to the solution

= 2 ln ∣ ln (x) ∣ + C

\frac{d y}{d x}, x y - 5 y^{2} = 6

\int x^{5} 4 + x^{6} d x

\int \frac{x ^{2}}{x ^{2} - 6 x + 10} d x

x \to 0 + lim (\frac{1}{5 x})

y = x^{3} - 2 x + 1, y = - 2 x, x = 1