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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

Add a constant to the solution

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

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