How do you integrate #f(x)=(x^2+x+1)(x-1)# using the product rule?

Question

Elijah Abram · Accepted Answer

There is no product rule for integration.

Isabella Ackerman · Answer

Jim H. is right when he says that there isn't a product rule for integration; however, there is an integration method known as Integration by Parts, which is so closely related to the product rule that the method can be derived from the product rule. I won't repeat the derivation in this post, but I will use the method to complete the integration.

NOTE: Since you asked for the use of the product rule, I will use the approach that is most similar to the product rule. Normally, I would not use integration by parts; instead, I would multiply, simplify, and then integrate each term.

The formula for integration by parts is:

#int udv = uv - intvdu#

To apply the formula to the given factors, we let: #u = (x-1)# and #dv = (x^2+x+1)dx# this implies that #du = 1 dx# and #v = (x^3/3+x^2/2+x)#

Changing the formula to use these four equations:

#int (x-1)(x^2+x+1)dx = (x-1)(x^3/3+x^2/2+x) - int(x^3/3+x^2/2+x)dx#

The remaining integral is done "term-by-term", using the power rule, #int x^n dx = 1/(n+1)x^(n+1) + C#:

#int (x-1)(x^2+x+1)dx = (x-1)(x^3/3+x^2/2+x) - (x^4/12+x^3/6+x^2/2)+ C#

Please contrast the following response with the result obtained by simply performing the multiplication:

#int (x-1)(x^2+x+1)dx = int x(x^2+x+1)-(x^2+x+1)dx#

#int (x-1)(x^2+x+1)dx = int x^3+x^2+x-x^2-x-1dx#

#int (x-1)(x^2+x+1)dx = int x^3-1dx#

#int (x-1)(x^2+x+1)dx = x^4/4-x+ C#

Unlike the other method, this one does not require the laborious simplification.

Elijah Abram · Answer

undefined To integrate $ f(x) = (x^2 + x + 1)(x - 1) $ using the product rule, follow these steps:

1. Identify the functions $ u $ and $ v $ where $ u = x^2 + x + 1 $ and $ v = x - 1 $.
2. Apply the product rule, which states that $ \int u \cdot v \, dx = u \cdot \int v \, dx - \int \left(\frac{du}{dx} \cdot \int v \, dx\right) \, dx $.
3. Find $ \frac{du}{dx} $ and $ \int v \, dx $.
4. Substitute these values into the formula and integrate.

So, the integral of $ f(x) $ using the product rule is:

$$ \int (x^2 + x + 1)(x - 1) \, dx = \frac{(x^2 + x + 1) \cdot (x^2/2 - x)}{2} - \int (2x + 1) \cdot \left(\frac{x^2}{2} - x\right) \, dx $$

You can simplify this expression further to obtain the final result.