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

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

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

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

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

Solution steps

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

Expand

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

= \int 4 x^{2} - 2 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 4 x^{2} d x - \int 2 x^{3} d x + \int x^{5} d x

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

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

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

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

Add a constant to the solution

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

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