What is the integral of (4-x^2)(2x-x^3) ?

The integral of (4-x^2)(2x-x^3) is 4x^2-(3x^4)/2+(x^6)/6+C

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integral of (4-x^2)(2x-x^3)

\int (4 - x^{2}) (2 x - x^{3}) d x

4 x^{2} - \frac{3 x ^{4}}{2} + \frac{x ^{6}}{6} + C

Solution steps

\int (4 - x^{2}) (2 x - x^{3}) d x

Expand

(4 - x^{2}) (2 x - x^{3}) : 8 x - 6 x^{3} + x^{5}

= \int 8 x - 6 x^{3} + x^{5} 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 8 x d x - \int 6 x^{3} d x + \int x^{5} d x

\int 8 x d x = 4 x^{2}

\int 6 x^{3} d x = \frac{3 x ^{4}}{2}

\int x^{5} d x = \frac{x ^{6}}{6}

= 4 x^{2} - \frac{3 x ^{4}}{2} + \frac{x ^{6}}{6}

Add a constant to the solution

= 4 x^{2} - \frac{3 x ^{4}}{2} + \frac{x ^{6}}{6} + C

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What is the integral of (4-x^2)(2x-x^3) ?
The integral of (4-x^2)(2x-x^3) is 4x^2-(3x^4)/2+(x^6)/6+C