Question

how we choose the function we let it equal u in integration by substitution ???
please any help

Answer 1

You chose the function that when you take its derivative, that derivative exists in the original problem.

For example if you have (3x^2)*(6x), and you are asked to integrate it, you can take 3x^2 to equal u,
\[u=3x ^{2}, du=6x dx\]

then you can substitute the 6xdx with just du.

Let me know if that doesnt make sense.

Answer 2

In principle, you can perform a substitution whenever you can completely replace the original variable.

Although in practice, not every substitution makes the integral easy to solve.

Answer 3

You have to be able to recognize something in the original integral itself. It is this:

if you see an integrand consisting of a product of functions, try to recognize if one of them is or might be the derivative of the other. If this is so, you let that function be u such that when you take du, you get the function you just recognized as being the derivative of the other that it multiplies. This allows you to totally change variables to u and du in the integral which is exactly like integrating xdx.

Example: \[\int\limits Sin(x)Cos(x) dx\]
We quickly recogniz that Cos(x) is the derivative of Sin(x). So, of course, then, you will allow

\[u = Sin(x) \] such that
\[du = Cos(x) dx\]

Substitue....--> and pretty simple from there isn't it?? :)

A useful help: the tricky part is recognizing what is what and you get this by simply famliarizing yourself with A BUNCH and BUNCH and BUNCH of 'important' integrals.