What is the integral of (x^2+5x^3-7)/(x^2) ?

The integral of (x^2+5x^3-7)/(x^2) is x+(5x^2)/2+7/x+C

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integral of (x^2+5x^3-7)/(x^2)

\int \frac{x ^{2} + 5 x ^{3} - 7}{x ^{2}} d x

x + \frac{5 x ^{2}}{2} + \frac{7}{x} + C

Solution steps

\int \frac{x ^{2} + 5 x ^{3} - 7}{x ^{2}} d x

Expand

\frac{x ^{2} + 5 x ^{3} - 7}{x ^{2}} : 1 + 5 x - \frac{7}{x ^{2}}

= \int 1 + 5 x - \frac{7}{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 5 x d x - \int \frac{7}{x ^{2}} d x

\int 1 d x = x

\int 5 x d x = \frac{5 x ^{2}}{2}

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

= x + \frac{5 x ^{2}}{2} - (- \frac{7}{x})

Simplify

= x + \frac{5 x ^{2}}{2} + \frac{7}{x}

Add a constant to the solution

= x + \frac{5 x ^{2}}{2} + \frac{7}{x} + C

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

x \to 2 - lim (x + 5)

\frac{5 x}{x + 3}

\int 2 + x d x

\frac{d}{d x} (\frac{25 - x ^{2}}{16 x})

What is the integral of (x^2+5x^3-7)/(x^2) ?
The integral of (x^2+5x^3-7)/(x^2) is x+(5x^2)/2+7/x+C