What is the integral of (\sqrt[3]{x}-4+5x)/(x^2) ?

The integral of (\sqrt[3]{x}-4+5x)/(x^2) is -3/(2x^{2/3)}+4/x+5ln|x|+C

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integral of (\sqrt[3]{x}-4+5x)/(x^2)

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

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

Solution steps

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

Expand

\frac{3 x - 4 + 5 x}{x ^{2}} : \frac{1}{x ^{\frac{5}{3}}} - \frac{4}{x ^{2}} + \frac{5}{x}

= \int \frac{1}{x ^{\frac{5}{3}}} - \frac{4}{x ^{2}} + \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 \frac{1}{x ^{\frac{5}{3}}} d x - \int \frac{4}{x ^{2}} d x + \int \frac{5}{x} d x

\int \frac{1}{x ^{\frac{5}{3}}} d x = - \frac{3}{2 x ^{\frac{2}{3}}}

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

\int \frac{5}{x} d x = 5 ln ∣ x ∣

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (\sqrt[3]{x}-4+5x)/(x^2) ?
The integral of (\sqrt[3]{x}-4+5x)/(x^2) is -3/(2x^{2/3)}+4/x+5ln|x|+C