What is the integral of (x^3+x^{-2})/(x^4-4x^{-1)} ?

The integral of (x^3+x^{-2})/(x^4-4x^{-1)} is 1/4 ln|x^4-4/x |+C

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integral of (x^3+x^{-2})/(x^4-4x^{-1)}

\int \frac{x ^{3} + x ^{- 2}}{x ^{4} - 4 x ^{- 1}} d x

\frac{1}{4} ln x^{4} - \frac{4}{x} + C

Solution steps

\int \frac{x ^{3} + x ^{- 2}}{x ^{4} - 4 x ^{- 1}} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

= \int \frac{1}{4 u} d u

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{4} ln ∣ u ∣

Substitute back

u = x^{4} - 4 x^{- 1}

= \frac{1}{4} ln x^{4} - 4 x^{- 1}

4 x^{- 1} = \frac{4}{x}

= \frac{1}{4} ln x^{4} - \frac{4}{x}

Add a constant to the solution

= \frac{1}{4} ln x^{4} - \frac{4}{x} + C

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What is the integral of (x^3+x^{-2})/(x^4-4x^{-1)} ?
The integral of (x^3+x^{-2})/(x^4-4x^{-1)} is 1/4 ln|x^4-4/x |+C