What is the integral of x^{-3}+7x^{-1} ?

The integral of x^{-3}+7x^{-1} is -1/(2x^2)+7ln|x|+C

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integral of x^{-3}+7x^{-1}

\int x^{- 3} + 7 x^{- 1} d x

- \frac{1}{2 x ^{2}} + 7 ln ∣ x ∣ + C

Solution steps

\int x^{- 3} + 7 x^{- 1} d x

Simplify

x^{- 3} + 7 x^{- 1} : \frac{1}{x ^{3}} + \frac{7}{x}

= \int \frac{1}{x ^{3}} + \frac{7}{x} 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 ^{3}} d x + \int \frac{7}{x} d x

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

\int \frac{7}{x} d x = 7 ln ∣ x ∣

= - \frac{1}{2 x ^{2}} + 7 ln ∣ x ∣

Add a constant to the solution

= - \frac{1}{2 x ^{2}} + 7 ln ∣ x ∣ + C

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What is the integral of x^{-3}+7x^{-1} ?
The integral of x^{-3}+7x^{-1} is -1/(2x^2)+7ln|x|+C