What is the integral of 1/(xln(x^9)) ?

The integral of 1/(xln(x^9)) is 1/9 ln(9|ln(x)|)+C

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integral of 1/(xln(x^9))

\int \frac{1}{x ln ( x ^{9} )} d x

\frac{1}{9} ln (9 ∣ ln (x) ∣) + C

Solution steps

\int \frac{1}{x ln ( x ^{9} )} d x

Apply u-substitution

: \int \frac{1}{9 u ln ( u )} d u

= \int \frac{1}{9 u ln ( u )} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int \frac{1}{u ln ( u )} d u

Apply u-substitution

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

= \frac{1}{9} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{9} ln ∣ v ∣

Substitute back

= \frac{1}{9} ln ln (x^{9})

Simplify

\frac{1}{9} ln ln (x^{9}) : \frac{1}{9} ln (9 ∣ ln (x) ∣)

= \frac{1}{9} ln (9 ∣ ln (x) ∣)

Add a constant to the solution

= \frac{1}{9} ln (9 ∣ ln (x) ∣) + C

\int_{0}^{1} \frac{2}{2 - x ^{2}} d x

y = 3 x - 1

f (x) = e^{x} (x + x x)

\frac{d y}{d x} = y - y^{2}

f (x) = - x^{3}, (1, - 1)

What is the integral of 1/(xln(x^9)) ?
The integral of 1/(xln(x^9)) is 1/9 ln(9|ln(x)|)+C