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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

\int \frac{7}{x} d x = 7 ln ∣ x ∣

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

Add a constant to the solution

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

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