What is the integral of ((3x^3-2x^5)/(x^2)) ?

The integral of ((3x^3-2x^5)/(x^2)) is -1/8 (-2x^2+3)^2+C

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integral of ((3x^3-2x^5)/(x^2))

\int (\frac{3 x ^{3} - 2 x ^{5}}{x ^{2}}) d x

- \frac{1}{8} (- 2 x^{2} + 3)^{2} + C

Solution steps

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

Simplify

\frac{3 x ^{3} - 2 x ^{5}}{x ^{2}} : x (- 2 x^{2} + 3)

= \int x (- 2 x^{2} + 3) d x

Apply u-substitution

: \int - \frac{u}{4} d u

= \int - \frac{u}{4} d u

Take the constant out:

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

= - \frac{1}{4} \cdot \int u d u

Apply the Power Rule

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

= - \frac{1}{4} \cdot \frac{u ^{2}}{2}

Substitute back

u = - 2 x^{2} + 3

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

Simplify

- \frac{1}{4} \cdot \frac{( - 2 x ^{2} + 3 ) ^{2}}{2} : - \frac{1}{8} (- 2 x^{2} + 3)^{2}

= - \frac{1}{8} (- 2 x^{2} + 3)^{2}

Add a constant to the solution

= - \frac{1}{8} (- 2 x^{2} + 3)^{2} + C

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What is the integral of ((3x^3-2x^5)/(x^2)) ?
The integral of ((3x^3-2x^5)/(x^2)) is -1/8 (-2x^2+3)^2+C