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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Add a constant to the solution

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

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