What is the integral of (1+x+sqrt(x))/(x^2) ?

The integral of (1+x+sqrt(x))/(x^2) is -1/x+ln|x|-2/(sqrt(x))+C

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integral of (1+x+sqrt(x))/(x^2)

\int \frac{1 + x + x}{x ^{2}} d x

- \frac{1}{x} + ln ∣ x ∣ - \frac{2}{x} + C

Solution steps

\int \frac{1 + x + x}{x ^{2}} d x

Expand

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

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

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= - \frac{1}{x} + ln ∣ x ∣ - \frac{2}{x}

Add a constant to the solution

= - \frac{1}{x} + ln ∣ x ∣ - \frac{2}{x} + C

d er i v a t i v e ln (\frac{x}{2})

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

\frac{d}{d x} (\frac{cos ( x )}{x ^{2} + tan ( x )})

x \to 7 + lim (\frac{8}{7 - x})

y^{^{'}} + y = \frac{1}{( cos ( 2 x ) ) ^{\frac{3}{2}}}

What is the integral of (1+x+sqrt(x))/(x^2) ?
The integral of (1+x+sqrt(x))/(x^2) is -1/x+ln|x|-2/(sqrt(x))+C