How do you integrate #f(x)=(2+x+2x^2-3x^3)(3-x+2x^2+2x^3)# using the product rule?

Answer 1

If #2+x+2x^2 -3x^3# =g(x) and #3-x+2x^2+2x^3#= h(x), then #int g(x)*h(x) dx=g(x) int h(x)dx - int d/dx g(x) int h(x)dx#. Accordingly,

#int g(x)*g(x) dx= (2+x+2x^2-3x^3)(3x-x^2/2 +2 x^3 /3 +2x^4 /4) -int (1+4x-9x^2)(3x-x^2/2 +2 x^3 /3 +2x^4 /4)dx #

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Answer 2Christopher Allers
To integrate ( f(x) = (2 + x + 2x^2 - 3x^3)(3 - x + 2x^2 + 2x^3) ) using the product rule, follow these steps:

Expand the product ( (2 + x + 2x^2 - 3x^3)(3 - x + 2x^2 + 2x^3) ) to get a polynomial expression.
Apply the product rule to integrate the expanded polynomial term by term, where each term is integrated separately.
Use the power rule for integration, integrating each term according to its power of ( x ).
Add the integrals of all terms together to get the final result.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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