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

The integral of ((sqrt(x)-1/2 x^2)^2)/(2x^2) is 1/2 ln|x|-1/3 x^{3/2}+(x^3)/(24)+C

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integral of ((sqrt(x)-1/2 x^2)^2)/(2x^2)

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

\frac{1}{2} ln ∣ x ∣ - \frac{1}{3} x^{\frac{3}{2}} + \frac{x ^{3}}{24} + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Expand

\frac{( x - \frac{1}{2} x ^{2} ) ^{2}}{x ^{2}} : \frac{1}{x} - x + \frac{x ^{2}}{4}

= \frac{1}{2} \cdot \int \frac{1}{x} - x + \frac{x ^{2}}{4} 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

= \frac{1}{2} (\int \frac{1}{x} d x - \int x d x + \int \frac{x ^{2}}{4} d x)

\int \frac{1}{x} d x = ln ∣ x ∣

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

\int \frac{x ^{2}}{4} d x = \frac{x ^{3}}{12}

= \frac{1}{2} (ln ∣ x ∣ - \frac{2}{3} x^{\frac{3}{2}} + \frac{x ^{3}}{12})

Simplify

\frac{1}{2} (ln ∣ x ∣ - \frac{2}{3} x^{\frac{3}{2}} + \frac{x ^{3}}{12}) : \frac{1}{2} ln ∣ x ∣ - \frac{1}{3} x^{\frac{3}{2}} + \frac{x ^{3}}{24}

= \frac{1}{2} ln ∣ x ∣ - \frac{1}{3} x^{\frac{3}{2}} + \frac{x ^{3}}{24}

Add a constant to the solution

= \frac{1}{2} ln ∣ x ∣ - \frac{1}{3} x^{\frac{3}{2}} + \frac{x ^{3}}{24} + C

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

What is the integral of ((sqrt(x)-1/2 x^2)^2)/(2x^2) ?
The integral of ((sqrt(x)-1/2 x^2)^2)/(2x^2) is 1/2 ln|x|-1/3 x^{3/2}+(x^3)/(24)+C