What is the integral of 1/(x^{8/9)(5+x^{1/9})} ?

The integral of 1/(x^{8/9)(5+x^{1/9})} is 9ln|5+x^{1/9}|+C

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integral of 1/(x^{8/9)(5+x^{1/9})}

\int \frac{1}{x ^{\frac{8}{9}} ( 5 + x ^{\frac{1}{9}} )} d x

9 ln 5 + x^{\frac{1}{9}} + C

Solution steps

\int \frac{1}{x ^{\frac{8}{9}} ( 5 + x ^{\frac{1}{9}} )} d x

Apply u-substitution

: \int \frac{9}{u} d u

= \int \frac{9}{u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 9 ln ∣ u ∣

Substitute back

u = 5 + x^{\frac{1}{9}}

= 9 ln 5 + x^{\frac{1}{9}}

Add a constant to the solution

= 9 ln 5 + x^{\frac{1}{9}} + C

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

2 x lo g_{4} (x)

\frac{\partial}{\partial x} (x^{2} + y^{2} + 3 z^{3})

\int \frac{2}{3} e^{3 x} x d x

\frac{d}{d x} (x^{4} + 5 x)

What is the integral of 1/(x^{8/9)(5+x^{1/9})} ?
The integral of 1/(x^{8/9)(5+x^{1/9})} is 9ln|5+x^{1/9}|+C