How do you integrate (indefinite) the product of two distinct terms, both containing x?

Well, the answer to this problem is sin(x^3)/3+C. When you differentiate this function, the chain rule comes into play, so the derivative of sin(x^3) is cos(x^3)(x^2)3, and you need to divide the entire function by 3 to cancel out the 3 at the end.

With experience you will be able to tell this just by looking, and recognizing that you have a function containing a second function, and the whole thing is multiplied by the derivative of that inside function (multiplied by a constant. this part is very important; if the derivative of the inside function is multiplied or divided by x or any other variable term, this technique will not work. But constants carry through, and indeed can be put outside of the integration/ differentiation process entirely, thus posing no problem). This is, of course, what you get when you use the chain rule to differentiate something. All you need to do is run the chain rule in reverse.

To formalize this process, use “u-substitution”, about which there is plenty of information on google. However, this only works for the situation I outlined above. There is an infinite number of multiple-part functions that require other methods to integrate, or are impossible to integrate explicitly by any means. Integration follows no hard and fast rules like differentiation, and is ultimately a matter of intuition, educated guesswork, and luck. So, good luck, and feel free to message me if you need more help.

Nice answer @Jayne.

@HeroicZach, there is no general “product rule” or “quotient rule” for integration. In addition to the substitution rule which is a method that sometimes works, there is also Integration by Parts that sometimes helps us integrate the product of two functions. I am not sure if it is covered in your class or not so I won’t comment further. But you can find a nice explanation here. Integration by Parts is generally considered an analogue of the Product Rule for differentiation.

P.S. Generally speaking you can very easily come up with functions that are “impossible” to integrate, which makes integration “harder” than differentiation. Example: exp(x^2).