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

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

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

\int \frac{2 x ^{4} - 3 x ^{3} + 5 x + 1}{x ^{3}} d x

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

Solution steps

\int \frac{2 x ^{4} - 3 x ^{3} + 5 x + 1}{x ^{3}} d x

Expand

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

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

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

\int 3 d x = 3 x

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

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

= x^{2} - 3 x - \frac{5}{x} - \frac{1}{2 x ^{2}}

Add a constant to the solution

= x^{2} - 3 x - \frac{5}{x} - \frac{1}{2 x ^{2}} + C

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