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

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

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

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

ln ∣ x ∣ + \frac{2}{5} x^{\frac{5}{2}} + C

Solution steps

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

Expand

x^{2} (\frac{1}{x ^{3}} + \frac{1}{x}) : \frac{1}{x} + x^{\frac{3}{2}}

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= ln ∣ x ∣ + \frac{2}{5} x^{\frac{5}{2}}

Add a constant to the solution

= ln ∣ x ∣ + \frac{2}{5} x^{\frac{5}{2}} + C

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\int_{0}^{10} 1

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

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