indefinite integral.

Question

Find the indefinite integral. Remember to include a "+ C" if appropriate.

∫((12x^3)-(12x))dx

Answer 1

Hello! For this answer, the first part of the answer is the actual answer. The second part is mathematical footnotes that explain my process and the general process to solve this type of problem. I hope this helps!

Sadly, Wyzant wasn't letting me attach my images itself, so I've had to attach this link that contains my answer.

https://imgur.com/a/DAY769V

Answer 2

Hi Femi,

Since we are integrating a polynomial function, we can simply rely on the following integration rule:

∫ axⁿ dx = (axⁿ⁺¹/n+1) + C

so for our example, integration gives:

(12x³⁺¹/(3+1)) - (12x¹⁺¹/(1+1)) + C

simplified:

3x⁴ - 6x² + C

The reason why a "+ C" is needed when we take an indefinite integral can be understood by remembering that integration is the "opposite" of differentiation (this is why you hear the term "antiderivative" used to describe integration). Now, when we differentiate a function, remember that any constant terms (which describe the vertical translation of a function such as for example y = x² - x + 50) become zero when we differentiate them. That means the following functions all have equivalent derivatives: y = x² - x + 1, y = x² - x + 2, y = x² - x + 3, ......). There is actually an infinite number of functions we can write here that satisfy this list. All of these have the derivative of y = 2x - 1. Now, when we return to taking an indefinite integral (which again is the "opposite" of differentiation), we have to acknowledge the fact that an infinite number of functions can be generated as antiderivative solutions. An indefinite integral of y = 2x - 1 in this example would generate that infinite list of functions that I listed above. This is why we have to add a "+ C" to our answer to an indefinite integral.