What is the integral of (2x^3-4x^4+7)/x ?

The integral of (2x^3-4x^4+7)/x is (2x^3)/3-x^4+7ln|x|+C

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integral of (2x^3-4x^4+7)/x

\int \frac{2 x ^{3} - 4 x ^{4} + 7}{x} d x

\frac{2 x ^{3}}{3} - x^{4} + 7 ln ∣ x ∣ + C

Solution steps

\int \frac{2 x ^{3} - 4 x ^{4} + 7}{x} d x

Expand

\frac{2 x ^{3} - 4 x ^{4} + 7}{x} : 2 x^{2} - 4 x^{3} + \frac{7}{x}

= \int 2 x^{2} - 4 x^{3} + \frac{7}{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 2 x^{2} d x - \int 4 x^{3} d x + \int \frac{7}{x} d x

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

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

\int \frac{7}{x} d x = 7 ln ∣ x ∣

= \frac{2 x ^{3}}{3} - x^{4} + 7 ln ∣ x ∣

Add a constant to the solution

= \frac{2 x ^{3}}{3} - x^{4} + 7 ln ∣ x ∣ + C

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What is the integral of (2x^3-4x^4+7)/x ?
The integral of (2x^3-4x^4+7)/x is (2x^3)/3-x^4+7ln|x|+C