What is the integral of ((x^3+x^2sin(x)-2))/(x^2) ?

The integral of ((x^3+x^2sin(x)-2))/(x^2) is (x^2)/2-cos(x)+2/x+C

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integral of ((x^3+x^2sin(x)-2))/(x^2)

\int \frac{( x ^{3} + x ^{2} sin ( x ) - 2 )}{x ^{2}} d x

\frac{x ^{2}}{2} - cos (x) + \frac{2}{x} + C

Solution steps

\int \frac{x ^{3} + x ^{2} sin ( x ) - 2}{x ^{2}} d x

Expand

\frac{x ^{3} + x ^{2} sin ( x ) - 2}{x ^{2}} : x + sin (x) - \frac{2}{x ^{2}}

= \int x + sin (x) - \frac{2}{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 x d x + \int sin (x) d x - \int \frac{2}{x ^{2}} d x

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

\int sin (x) d x = - cos (x)

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

= \frac{x ^{2}}{2} - cos (x) - (- \frac{2}{x})

Simplify

= \frac{x ^{2}}{2} - cos (x) + \frac{2}{x}

Add a constant to the solution

= \frac{x ^{2}}{2} - cos (x) + \frac{2}{x} + C

\int \frac{3 x ^{2} - 5 x + 1}{x ^{3} - x ^{2} - 2 x + 30} d x

\int - 195 sin (7.8 t) d t

\int x (x + 1)^{7} d x

\int \frac{1 ln ( x )}{x} d x

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

What is the integral of ((x^3+x^2sin(x)-2))/(x^2) ?
The integral of ((x^3+x^2sin(x)-2))/(x^2) is (x^2)/2-cos(x)+2/x+C