Substitution method of Integration, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com PDF Download

Integration by Substitution

"Integration by Substitution" (also called "u-substitution") is a method to find an integral, but only when it can be set up in a special way.

The first and most vital step is to be able to write our integral in this form:

Note that we have g(x) and its derivative g'(x)

Like in this example:  

Here f=cos, and we have g=x² and its derivative of 2x
 This integral is good to go!

When our integral is set up like that, we can do this substitution:

Then we can integrate f(u), and finish by putting g(x) back as u.

Like this:

Example: 

We know (from above) that it is in the right form to do the substitution:

 Now integrate:
    |
 And finally put u=x² back again:
 sin(x²) + C

 worked out really nicely! (Well, I knew it would.)

This method only works on some integrals of course, and it may need rearranging:
Example: 

Oh no! It is 6x, not 2x. Our perfect setup is gone.

Never fear! Just rearrange the integral like this:

(We can pull constant multipliers outside the integration, see Rules of Integration.)

Then go ahead as before:

 Now put u=x² back again:
 3 sin(x²) + C
 Done!
 Now we are ready for a slightly harder example:
 Example:  
 Let me see ... the derivative of x²+1 is 2x ... so how about we rearrange it like this:

 Then we have:

 Then integrate:

Now put u=x²+1 back again:
 ½ ln(x²+1) + C
 And how about this one:
Example:  

Let me see ... the derivative of x+1 is ... well it is simply 1.

So we can have this:

Then we have:

 Then integrate:

 Now put u=x+1 back again:

In Summary

• When we can put an integral in this form:
  
• Then we can make u=g(x) and integrate ∫f(u) du
• And finish up by re-inserting g(x) where u is.