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

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

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

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

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

Solution steps

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

Expand

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

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

\int 3 x^{2} d x = x^{3}

\int 4 x d x = 2 x^{2}

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

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

Add a constant to the solution

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

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