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

\frac{2}{x}

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

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

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

\int \frac{x ^{2}}{( x - 1 ) ^{100}} 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