What is the integral of (4+sqrt(x))/(x^3) ?

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

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integral of (4+sqrt(x))/(x^3)

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

- \frac{2}{x ^{2}} - \frac{2}{3 x ^{\frac{3}{2}}} + C

Solution steps

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

Expand

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

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

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

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

= - \frac{2}{x ^{2}} - \frac{2}{3 x ^{\frac{3}{2}}}

Add a constant to the solution

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

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