What is the integral of (2x^5-x^2+1)/(x^4) ?

The integral of (2x^5-x^2+1)/(x^4) is x^2+1/x-1/(3x^3)+C

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integral of (2x^5-x^2+1)/(x^4)

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

x^{2} + \frac{1}{x} - \frac{1}{3 x ^{3}} + C

Solution steps

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

Expand

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

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

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

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

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

= x^{2} - (- \frac{1}{x}) - \frac{1}{3 x ^{3}}

Simplify

= x^{2} + \frac{1}{x} - \frac{1}{3 x ^{3}}

Add a constant to the solution

= x^{2} + \frac{1}{x} - \frac{1}{3 x ^{3}} + C

What is the integral of (2x^5-x^2+1)/(x^4) ?
The integral of (2x^5-x^2+1)/(x^4) is x^2+1/x-1/(3x^3)+C