What is the integral of (x^2+1)/(x^{5/2)} ?

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

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integral of (x^2+1)/(x^{5/2)}

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

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

Solution steps

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

Expand

\frac{x ^{2} + 1}{x ^{\frac{5}{2}}} : \frac{1}{x ^{\frac{1}{2}}} + \frac{1}{x ^{\frac{5}{2}}}

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

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

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

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

Add a constant to the solution

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

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