What is the integral of (-2x^5+5x^4+x^3+6)/(x^4) ?

The integral of (-2x^5+5x^4+x^3+6)/(x^4) is -x^2+5x+ln|x|-2/(x^3)+C

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integral of (-2x^5+5x^4+x^3+6)/(x^4)

\int \frac{- 2 x ^{5} + 5 x ^{4} + x ^{3} + 6}{x ^{4}} d x

- x^{2} + 5 x + ln ∣ x ∣ - \frac{2}{x ^{3}} + C

Solution steps

\int \frac{- 2 x ^{5} + 5 x ^{4} + x ^{3} + 6}{x ^{4}} d x

Expand

\frac{- 2 x ^{5} + 5 x ^{4} + x ^{3} + 6}{x ^{4}} : - 2 x + 5 + \frac{1}{x} + \frac{6}{x ^{4}}

= \int - 2 x + 5 + \frac{1}{x} + \frac{6}{x ^{4}} 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 2 x d x + \int 5 d x + \int \frac{1}{x} d x + \int \frac{6}{x ^{4}} d x

\int 2 x d x = x^{2}

\int 5 d x = 5 x

\int \frac{1}{x} d x = ln ∣ x ∣

\int \frac{6}{x ^{4}} d x = - \frac{2}{x ^{3}}

= - x^{2} + 5 x + ln ∣ x ∣ - \frac{2}{x ^{3}}

Add a constant to the solution

= - x^{2} + 5 x + ln ∣ x ∣ - \frac{2}{x ^{3}} + C

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What is the integral of (-2x^5+5x^4+x^3+6)/(x^4) ?
The integral of (-2x^5+5x^4+x^3+6)/(x^4) is -x^2+5x+ln|x|-2/(x^3)+C