What is the integral of (x^2+3sqrt(x)-4/(x^2)+2+x) ?

The integral of (x^2+3sqrt(x)-4/(x^2)+2+x) is (x^3)/3+2x^{3/2}+4/x+2x+(x^2)/2+C

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integral of (x^2+3sqrt(x)-4/(x^2)+2+x)

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

\frac{x ^{3}}{3} + 2 x^{\frac{3}{2}} + \frac{4}{x} + 2 x + \frac{x ^{2}}{2} + C

Solution steps

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

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

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

\int \frac{4}{x ^{2}} d x = - \frac{4}{x}

\int 2 d x = 2 x

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

= \frac{x ^{3}}{3} + 2 x^{\frac{3}{2}} - (- \frac{4}{x}) + 2 x + \frac{x ^{2}}{2}

Simplify

= \frac{x ^{3}}{3} + 2 x^{\frac{3}{2}} + \frac{4}{x} + 2 x + \frac{x ^{2}}{2}

Add a constant to the solution

= \frac{x ^{3}}{3} + 2 x^{\frac{3}{2}} + \frac{4}{x} + 2 x + \frac{x ^{2}}{2} + C

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What is the integral of (x^2+3sqrt(x)-4/(x^2)+2+x) ?
The integral of (x^2+3sqrt(x)-4/(x^2)+2+x) is (x^3)/3+2x^{3/2}+4/x+2x+(x^2)/2+C