For this, we observe that the term outside the bracket is actually the derivative of the expression inside the bracket

do we always use the u substitution method for this kind of integration of 2product functions?

im actually confused as i nvr seen this type of question in olevel syllabus before. also not sure what ia going on in the person's answer

Do you still remember chain rule from the past?

A nice thing to remember is that we differentiate expressions of the form [f(x)]^n, we will always have to differentiate the term inside later on.

We will get n times [f(x)]^(n - 1) times f'(x). In other words, we have the original function (with one power lower) and the differentiated function attached outside it as well.

If we reverse the order, as long as we have the f'(x) being multiplied to the expression, we can integrate them together.

In other words, integrating expressions of the form f'(x) times [f(x)]^n gets us

[f(x)]^(n + 1) divided by (n + 1), as long as n is not equal to -1.

We are essentially reversing the order of the differentiation.

In this example, the expression 4x³ is just nice the derivative of (x⁴ - 1).

Then, we can integrate this expression as a whole. Note that integrating (x⁴ - 1)^(1/2) on its own will not be that easy. The 4x³ makes the integration possible.

We will get (x⁴ - 1)^(3/2), whole expression divided by (3/2), plus the standard constant c.

ohh but for the process of solving this qn, must we always use the u substitution method? or is there an easier way?

I wouldn’t call this integration by substitution. In fact, this is just a reversal of the chain rule.

ohh, but actually to solve this kind of qn integrating f'(x)f(x) , i can jst ignore the f'(x) and focus on the f(x) right?

like just integrating the f(x) itself by doing f(x) ^n+1/n+1?

but i also noticed there are some qns that isnt related to the chain rule (not integrating f'(x) f(x))

Of course, there are different types of integrals in the JC syllabus and poly calculus courses. The ones in the O Level are meant to be “introductory” to let students have a basic feel of how the basic differentiation rules work (when y can be expressed in terms of x) and their basic applications.

There are further rules to know (such as the derivative of the inverse of sine) which are more advanced and are therefore not in the O Level syllabus. Derivatives are required for any function, even those in which we cannot express y in terms of x. These are also not covered in the O Levels, but covered in A Levels and poly calculus courses. There are further applications too, such as integration by parts, integration by substitution and volume of revolution.

In the case of f’(x) multiplied by [f(x)]^n, we have to first “suspect” that this is the format, before inspecting that the term outside f’(x) is indeed the derivative of f(x). If yes, then the function integrates to [f(x)]^(n + 1), whole thing divided by (n + 1) if n is not equal to -1 or the function integrates to ln |f(x) | where the expression | | indicates a modulus/absolute function. Note that this | | was not seen at the O Level integration as well.

Ensure that the format is correct, because if the outside term is not f’(x) (as is the case for your other question posted), then the idea does not work.