Substitution and integration by parts are both integration techniques that can be used to evaluate integrals. The choice between the two depends on the form of the integral. Generally, substitution is useful when the integral contains a function and its derivative, while integration by parts is useful when the integral contains a product of functions. It is important to use your judgement and try both techniques if necessary.
Choosing the right substitution is key to successfully evaluating an integral. A good strategy is to look for a function in the integral that resembles a basic derivative or an antiderivative. Sometimes it is helpful to let u be the entire expression inside the integral, or to use trigonometric substitutions when dealing with trigonometric functions.
The substitution rule for integrals is given by: ∫f(g(x))g'(x) dx = ∫f(u) du. This means that when using substitution, we replace the variable x with a new variable u inside the integral, and also replace dx with du. This allows us to evaluate the integral in terms of u instead of x.
Yes, it is possible to use both techniques in the same integral. This is known as the method of "double substitution" or "substitution followed by integration by parts". It can be useful when the integral contains both a product of functions and a function and its derivative.
There is no specific order in which these techniques should be applied. As mentioned before, it is important to use your judgement and try both techniques if necessary. Sometimes it may be helpful to use one technique, and then switch to the other if the integral becomes more manageable.