How do you integrate #f(x)=(-3+3x+3x^2-3x^3)(3-3x-x^2+3x^3+2x^4)# using the product rule?

Question

Samantha Adkison · Accepted Answer

#-3/4x^8-3/7x^7+3x^6+9/5x^5-15/2x^4+x^3+9x^2-9x+C#

The product rule only applies to derivatives; integrals are not covered by it.

In this case, we have to just expand the entire thing: #(-3+3x+3x^2-3x^3)(3-3x-x^2+3x^3+2x^4)=#

#=-9+9x+3x^2-9x^3-6x^4+9x-9x^2-3x^3+9x^4+6x^5+9x^2-9x^3-3x^4+9x^5+6x^6-9x^3+9x^4+3x^5-9x^6-6x^7#

Combining like terms, we get: #-6x^7-3x^6+18x^5+9x^4-30x^3+3x^2+18x-9#

We can then integrate using the reverse power rule, which says: #int\ x^n\ dx=x^(n+1)/(n+1)+C#

In our case, we get: #int\ -6x^7-3x^6+18x^5+9x^4-30x^3+3x^2+18x-9\ dx=#

#=-6/8x^8-3/7x^7+18/6x^6+9/5x^5-30/4x^4+3/3x^3+18/2x^2-9x+C=#

Simplifying, we get: #=-3/4x^8-3/7x^7+3x^6+9/5x^5-15/2x^4+x^3+9x^2-9x+C#

Paisley Johnson · Answer

undefined To integrate $ f(x) = (-3 + 3x + 3x^2 - 3x^3)(3 - 3x - x^2 + 3x^3 + 2x^4) $ using the product rule, follow these steps:

1. Expand the product $ f(x) $ to obtain a polynomial expression.
2. Apply the power rule to integrate each term individually.
3. Add the integrated terms together to find the final result.

Here's the detailed solution:

1. Expand $ f(x) $:
$$ f(x) = (-3)(3) + (-3)(-3x) + (-3)(-x^2) + (-3)(3x^3) + (-3)(2x^4) + (3x)(3) + (3x)(-3x) + (3x)(-x^2) + (3x)(3x^3) + (3x)(2x^4) + (3x^2)(3) + (3x^2)(-3x) + (3x^2)(-x^2) + (3x^2)(3x^3) + (3x^2)(2x^4) + (-3x^3)(3) + (-3x^3)(-3x) + (-3x^3)(-x^2) + (-3x^3)(3x^3) + (-3x^3)(2x^4) $$
2. Simplify each term and integrate using the power rule:
$$ F(x) = \frac{-3x^2}{2} + \frac{3x^3}{2} - x^3 + \frac{3x^5}{5} - \frac{x^6}{3} + \frac{3x^3}{2} - \frac{3x^4}{2} + x^5 - \frac{3x^7}{7} + \frac{2x^8}{4} + \frac{3x^4}{2} - \frac{3x^5}{2} + x^6 - \frac{3x^8}{8} + \frac{2x^9}{5} - \frac{3x^5}{2} + \frac{3x^6}{2} - x^7 + \frac{3x^9}{9} - \frac{2x^{10}}{10} $$
3. Combine like terms to get the final result:
$$ F(x) = \frac{-3x^2}{2} + \frac{3x^3}{2} - x^3 + \frac{3x^5}{5} - \frac{x^6}{3} + \frac{3x^3}{2} - \frac{3x^4}{2} + x^5 - \frac{3x^7}{7} + \frac{2x^8}{4} + \frac{3x^4}{2} - \frac{3x^5}{2} + x^6 - \frac{3x^8}{8} + \frac{2x^9}{5} - \frac{3x^5}{2} + \frac{3x^6}{2} - x^7 + \frac{3x^9}{9} - \frac{2x^{10}}{10} $$