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

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

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

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

\frac{x ^{4}}{4} + \frac{x ^{3}}{3} + 2 x^{2} - 5 ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

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

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

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + 2 x^{2} - 5 ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{4}}{4} + \frac{x ^{3}}{3} + 2 x^{2} - 5 ln ∣ x ∣ + C

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