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

The integral of (x-1/(x^2))(x^2+2/x) is ((x^3+2)^2)/(4x^2)+C

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

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

\frac{( x ^{3} + 2 ) ^{2}}{4 x ^{2}} + C

Solution steps

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

Apply u-substitution

: \int \frac{u}{2} d u

= \int \frac{u}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= \frac{1}{2} \cdot \frac{u ^{2}}{2}

Substitute back

u = x^{2} + \frac{2}{x}

= \frac{1}{2} \cdot \frac{( x ^{2} + \frac{2}{x} ) ^{2}}{2}

Simplify

\frac{1}{2} \cdot \frac{( x ^{2} + \frac{2}{x} ) ^{2}}{2} : \frac{( x ^{3} + 2 ) ^{2}}{4 x ^{2}}

= \frac{( x ^{3} + 2 ) ^{2}}{4 x ^{2}}

Add a constant to the solution

= \frac{( x ^{3} + 2 ) ^{2}}{4 x ^{2}} + C

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What is the integral of (x-1/(x^2))(x^2+2/x) ?
The integral of (x-1/(x^2))(x^2+2/x) is ((x^3+2)^2)/(4x^2)+C