MAT133Y5 Chapter Notes - Chapter 7: Differential Calculus, Antiderivative
CHAPTER 7- INTEGRATION TECHNIQUES
SUMMARY
Having seen that integration and differentiation are essentially inverses,
we would like to develop some techniques and rules form computing
integrals. It should be unsurprising that those rules will arise as the
“inverse” operations of the rules obtained from differential calculus.
Unfortunately, the majority of times nature will conspire against us and
not write our integrand so plainly as f′(g(x))g′(x). Hence we develop
some techniques to make life simpler. Our strategy is as follows:
1. Look to see if we can find the occurrence of a function and its
derivative. In (7.1) above, we are looking for the function g(x),
since it occurs in the argument of f(x) and its derivative appears as
g′(x).
2. Define a new variable, u = g(x) so that du = g′(x). We often write
this second equation as dx du = g′(x) dx.
3. Replace all x dependencies with u dependencies
4. Using the fundamental theorem of calculus, evaluate our new
integral:
5. We now have our solution, but it is in terms of the variable u. This
is not a problem since we know that u = g(x), so we just make
this substitution to get our final solution
When dealing with definite integrals, we adhere to the same process
as indefinite integrals, but we must also accommodate the lower and
upper bounds of integration. Let f be a continuous function with anti-
derivative F, while g is continuously differentiable.
Once again consider the case when we are integrating the function
find more resources at oneclass.com
find more resources at oneclass.com
64
MAT133Y5 Full Course Notes
Verified Note
64 documents
Document Summary
Having seen that integration and differentiation are essentially inverses, we would like to develop some techniques and rules form computing integrals. It should be unsurprising that those rules will arise as the. Inverse operations of the rules obtained from differential calculus. Unfortunately, the majority of times nature will conspire against us and not write our integrand so plainly as f (g(x))g (x). Hence we develop some techniques to make life simpler. Look to see if we can find the occurrence of a function and its derivative. In (7. 1) above, we are looking for the function g(x), since it occurs in the argument of f(x) and its derivative appears as g (x): define a new variable, u = g(x) so that du = g (x). This is not a problem since we know that u = g(x), so we just make this substitution to get our final solution.
