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Techniques of Integration
The reverse chain rule, integration by substitution, integration by parts, additional strategies
The reverse chain rule, integration by substitution, integration by parts, additional strategies
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If ​F(x)=∫f(x)dx, then
Integrating a composition of functions ​f(g(x))​ requires us to divide by ​g′(x), so it is easier to find the anti-derivative of anything of the form ​g′(x)f′(g(x))​ by first dividing by ​g′(x).
In symbols, we use the known fact
and let ​u=g(x), giving us
an integral we can solve more easily:
Then, we substitute ​g(x)​ back in to get our desired result of ​kf(g(x))+C.
When we make a substitution in a definite integral in the form
we need to remember that the bounds are from ​x=a​ to ​x=b:
We then have two choices:
Plug ​x=a​ and ​x=b​ into ​u​ to find the bounds in terms of ​u.
Plug ​u(x)​ back in and use the bounds ​a→b.