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

The integral of x^2(sin(2)x^3+4x) is 1/6 sin(2)x^6+x^4+C

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

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

\frac{1}{6} sin (2) x^{6} + x^{4} + C

Solution steps

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

Expand

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

= \int sin (2) x^{5} + 4 x^{3} 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 sin (2) x^{5} d x + \int 4 x^{3} d x

\int sin (2) x^{5} d x = \frac{1}{6} sin (2) x^{6}

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

= \frac{1}{6} sin (2) x^{6} + x^{4}

Add a constant to the solution

= \frac{1}{6} sin (2) x^{6} + x^{4} + C

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What is the integral of x^2(sin(2)x^3+4x) ?
The integral of x^2(sin(2)x^3+4x) is 1/6 sin(2)x^6+x^4+C