What is the integral of (12x^{-4}-4x^{-3}+9x^{-1}+3) ?

The integral of (12x^{-4}-4x^{-3}+9x^{-1}+3) is -4/(x^3)+2/(x^2)+9ln|x|+3x+C

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integral of (12x^{-4}-4x^{-3}+9x^{-1}+3)

\int (12 x^{- 4} - 4 x^{- 3} + 9 x^{- 1} + 3) d x

- \frac{4}{x ^{3}} + \frac{2}{x ^{2}} + 9 ln ∣ x ∣ + 3 x + C

Solution steps

\int 12 x^{- 4} - 4 x^{- 3} + 9 x^{- 1} + 3 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 12 x^{- 4} d x - \int 4 x^{- 3} d x + \int 9 x^{- 1} d x + \int 3 d x

\int 12 x^{- 4} d x = - \frac{4}{x ^{3}}

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

\int 9 x^{- 1} d x = 9 ln ∣ x ∣

\int 3 d x = 3 x

= - \frac{4}{x ^{3}} - (- \frac{2}{x ^{2}}) + 9 ln ∣ x ∣ + 3 x

Simplify

= - \frac{4}{x ^{3}} + \frac{2}{x ^{2}} + 9 ln ∣ x ∣ + 3 x

Add a constant to the solution

= - \frac{4}{x ^{3}} + \frac{2}{x ^{2}} + 9 ln ∣ x ∣ + 3 x + C

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What is the integral of (12x^{-4}-4x^{-3}+9x^{-1}+3) ?
The integral of (12x^{-4}-4x^{-3}+9x^{-1}+3) is -4/(x^3)+2/(x^2)+9ln|x|+3x+C