EVALUATING INTEGRALS - SUBStitutiON EVALUATING INTEGRALS USING THE SUBSTITUTION RULE (PART I) In previous section, we've dealt with definite and indefinite integrals that can be solved by applying the Fundamental Theorems. However, there's a limit; there are more complex integrals which can't be solved directly with the Fundamental Theorem. Take a good look at this indefinite integral:
∫
4 (1 + 2x)3
dx
Do you think you can solve this integral directly using FTC? Obviously not! You'll realize that FTC isn't exactly well “equipped” to handle relatively complex integrals like the one above (at least, not in this form). So how do we go about it? First, understand the problem: in this form, the integral is too complex for the FTC to handle. So, we reduce it to a simpler form. After that, FTC can now be applied. The first step to be taken is to make a substitution. For starters, this step will make the integral look less overwhelming. The integrand here is the function
f (x)
4 (1 + 2x)3
=
Using a new variable u (the letter u is more commonly used for substitutions), we let
u = 1 + 2x
[STAGE 1]
So that the integral becomes
∫
4 dx u3
∫ 4u
OR
–3
dx
[STAGE 2]
Now we're getting somewhere! However, something is out of place in the integral
∫ 4u
–3
dx
At this point, we are integrating with respect to the new variable u, thus the symbol dx has to be replaced. We do this by making a second substitution by making a reference to the first substitution in stage 1:
u = 1 + 2x We differentiate u with respect to x:
du/dx = 2
[STAGE 3]
Next, make dx the subject:
dx = ½ du
[STAGE 4]
Then we go back to stage 2: replace dx with ½ du:
∫ (4u
–3
) × ½du
From the properties of integrals, ½ is a constant. Therefore we rewrite the integral:
½
∫ (4u
–3
) du
[STAGE 5]
At this point, you should see how this method works. In this example, by introducing a new variable u, we have reduced