Partial fractions in integration

Okay kiddo, so let's say you have a big fraction like 3x/(x^2 - 4). That looks pretty complicated, right? But we can make it simpler by splitting it up into smaller fractions.

Let me explain it with an example. Suppose you have some chocolates that are cut into pieces. You have 6 pieces of chocolates and you want to divide them equally between 2 children. How many pieces will each child get? They will get 3 pieces each, right?

Similarly, we can divide the big fraction 3x/(x^2 - 4) into 2 smaller fractions like this: 3x/(x^2 - 4) = A/(x-2) + B/(x+2).

Here, A and B are two special numbers that we need to find out. We call them "constants". They don't change even if you change the value of x.

So now we have two fractions: A/(x-2) and B/(x+2). If we add them, we get the original fraction 3x/(x^2 - 4).

But how do we find the values of A and B? That's where the magic of partial fractions comes in. We can use some tricks to find the values of A and B.

One way is to multiply both sides of the equation with the denominator (x^2 - 4). That gives us:

3x = A(x+2) + B(x-2)

Now we can substitute some values of x to find the values of A and B. For example, if we substitute x=2, we get:

6A = 6

So, A = 1.

Similarly, if we substitute x=-2, we get:

-6B = -6

So, B = 1.

Now we know the values of A and B. We can substitute them back into the original equation and simplify it:

3x/(x^2 - 4) = 1/(x-2) + 1/(x+2)

And that's partial fractions in integration. It's a way to split up big fractions into smaller ones so that we can solve them easily.