What is the integral of (x^2-x^5+sqrt(x))/(x^5) ?

The integral of (x^2-x^5+sqrt(x))/(x^5) is -1/(2x^2)-x-2/(7x^{7/2)}+C

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integral of (x^2-x^5+sqrt(x))/(x^5)

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

- \frac{1}{2 x ^{2}} - x - \frac{2}{7 x ^{\frac{7}{2}}} + C

Solution steps

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

Expand

\frac{x ^{2} - x ^{5} + x}{x ^{5}} : \frac{1}{x ^{3}} - 1 + \frac{1}{x ^{\frac{9}{2}}}

= \int \frac{1}{x ^{3}} - 1 + \frac{1}{x ^{\frac{9}{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{1}{x ^{3}} d x - \int 1 d x + \int \frac{1}{x ^{\frac{9}{2}}} d x

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

\int 1 d x = x

\int \frac{1}{x ^{\frac{9}{2}}} d x = - \frac{2}{7 x ^{\frac{7}{2}}}

= - \frac{1}{2 x ^{2}} - x - \frac{2}{7 x ^{\frac{7}{2}}}

Add a constant to the solution

= - \frac{1}{2 x ^{2}} - x - \frac{2}{7 x ^{\frac{7}{2}}} + C

\int \frac{3 e ^{3 x}}{x + 10} d x

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\int e^{- 3 x} - 3 x d x

\int \frac{1}{( 4 x + 5 ) ^{3}} d x

\int arccsc (2 x) d x

What is the integral of (x^2-x^5+sqrt(x))/(x^5) ?
The integral of (x^2-x^5+sqrt(x))/(x^5) is -1/(2x^2)-x-2/(7x^{7/2)}+C