What is the integral of sqrt(x)+1/(x^2)+5/(x^3) ?

The integral of sqrt(x)+1/(x^2)+5/(x^3) is 2/3 x^{3/2}-1/x-5/(2x^2)+C

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integral of sqrt(x)+1/(x^2)+5/(x^3)

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

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

Solution steps

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

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

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

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

= \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{x} - \frac{5}{2 x ^{2}}

Add a constant to the solution

= \frac{2}{3} x^{\frac{3}{2}} - \frac{1}{x} - \frac{5}{2 x ^{2}} + C

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What is the integral of sqrt(x)+1/(x^2)+5/(x^3) ?
The integral of sqrt(x)+1/(x^2)+5/(x^3) is 2/3 x^{3/2}-1/x-5/(2x^2)+C