What is the integral of 5x^{-3}+8x^{-2}+2x^{-1} ?

The integral of 5x^{-3}+8x^{-2}+2x^{-1} is -5/(2x^2)-8/x+2ln|x|+C

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integral of 5x^{-3}+8x^{-2}+2x^{-1}

\int 5 x^{- 3} + 8 x^{- 2} + 2 x^{- 1} d x

- \frac{5}{2 x ^{2}} - \frac{8}{x} + 2 ln ∣ x ∣ + C

Solution steps

\int 5 x^{- 3} + 8 x^{- 2} + 2 x^{- 1} d x

Simplify

5 x^{- 3} + 8 x^{- 2} + 2 x^{- 1} : \frac{5}{x ^{3}} + \frac{8}{x ^{2}} + \frac{2}{x}

= \int \frac{5}{x ^{3}} + \frac{8}{x ^{2}} + \frac{2}{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{5}{x ^{3}} d x + \int \frac{8}{x ^{2}} d x + \int \frac{2}{x} d x

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

\int \frac{8}{x ^{2}} d x = - \frac{8}{x}

\int \frac{2}{x} d x = 2 ln ∣ x ∣

= - \frac{5}{2 x ^{2}} - \frac{8}{x} + 2 ln ∣ x ∣

Add a constant to the solution

= - \frac{5}{2 x ^{2}} - \frac{8}{x} + 2 ln ∣ x ∣ + C

\int e^{- x} sin (x) + e^{- x} cos (x) d x

\int cos (x + 3) d x

\int - \frac{1}{8} cos (8 x) d x

\int \frac{y + 2}{y - 1} d y

What is the integral of 5x^{-3}+8x^{-2}+2x^{-1} ?
The integral of 5x^{-3}+8x^{-2}+2x^{-1} is -5/(2x^2)-8/x+2ln|x|+C