Example 1. (Chain rule) Evaluate
Algebrically reading the equation it looks like "multiplying and divising the same thing to the entire function like rationalization and transforming complex number to standard form.
Apply chain rule to a composite function:
Now integrate both sides once:
We can transform the above equation to another common representation for easier reading
Theorem 1. Integration by substitution
For those who were not familiar with substitution, you may follow this flowchart:
STEP 1: Identify f'(x) and g(f(x)). In most case in college level this is clear because g(f(x)) is supposed to be the ONLY composite function that can't be easily decomposed.
Example 2a. Identifying composite function Decide which one should be the composite function: i)
For i) and ii)this is clear that there's only one function has hard to break down, that is sin (x^2) and e^(x^3) respectively. For iii), first we know that it's a polynomial, and we know that
Theorem 2. Exceptional case of substitution
STEP 2: Write
Note that the constant matters so that we may have to put a constant multiplying the integrand.
Example 2b. Conduct step 2 for the three examples in example 2a.
i)
ii)
iii)
STEP 3: Finish the integration. This one should be simple since the function is in
Example 2c. Finish the integration as shown in example 2b.
i)
ii)
iii)
Note that we have verified power rule in example (iii).
Example 3. (Verification of power rule) Show that
For
Sometimes the function will not be as trivial as it shown above. The first extention is about constants sticking in the function.
Theorem 3. (Derivatives on function with constants)
This is a fact that has been proved in part II, but the one we have to use is:
Corollary 1.
WARNING. Note that
Therefore it allows us to ignore to do substitution without trouble from constant.
Example 4. Evaluate
We have
Another exception from substitution is a linear f(x).
Let
Consider
Theorem 4. (Linear composite functions)
Example 5. Evaluate the following integral: i)
i) We have
ii)
Now we end up this section with two harder example:
Example 6. (Complicated f(x)) Evaluate the integral
Noticing that
If there's an alternative method which is by partial fraction which we may discuss later. Noticing that
Example 7. (Multiple composite functions) Evaluate the integral
In this case we have no choice but to use substitution twice.
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