What is the integral of (x^{-3})/(x^{-2)+5} ?

The integral of (x^{-3})/(x^{-2)+5} is ln|x|-1/2 ln|5x^2+1|+C

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integral of (x^{-3})/(x^{-2)+5}

\int \frac{x ^{- 3}}{x ^{- 2} + 5} d x

ln ∣ x ∣ - \frac{1}{2} ln 5 x^{2} + 1 + C

Solution steps

\int \frac{x ^{- 3}}{x ^{- 2} + 5} d x

Simplify

\frac{x ^{- 3}}{x ^{- 2} + 5} : \frac{1}{x ( 5 x ^{2} + 1 )}

= \int \frac{1}{x ( 5 x ^{2} + 1 )} d x

Take the partial fraction of

\frac{1}{x ( 5 x ^{2} + 1 )} : \frac{1}{x} - \frac{5 x}{5 x ^{2} + 1}

= \int \frac{1}{x} - \frac{5 x}{5 x ^{2} + 1} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x} d x - \int \frac{5 x}{5 x ^{2} + 1} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{5 x}{5 x ^{2} + 1} d x = \frac{1}{2} ln 5 x^{2} + 1

= ln ∣ x ∣ - \frac{1}{2} ln 5 x^{2} + 1

Add a constant to the solution

= ln ∣ x ∣ - \frac{1}{2} ln 5 x^{2} + 1 + C

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What is the integral of (x^{-3})/(x^{-2)+5} ?
The integral of (x^{-3})/(x^{-2)+5} is ln|x|-1/2 ln|5x^2+1|+C