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

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

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

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

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

Solution steps

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

Expand

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

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

\int 1 d x = x

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

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

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

Add a constant to the solution

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

\int \frac{1}{9 + 16 x - 4 x ^{2}} d x

\int \frac{x ^{2}}{( x ^{2} + 1 ) ^{5}} d x

\int 45 (1 + x^{2}) (7 + 15 x + 5 x^{3})^{10} d x

\int \frac{x ^{3} - 4 x + 6}{x + 7} d x

\int (2 x)^{2} d x

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