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Integration techniques
We know that a definite integral can represent area and we've seen how this is connected to the idea of an anti-derivative through the Fundamental Theorem of Calculus (which is why we also use the integration symbol for anti-derivatives as well). Now, we'll build out our toolkit for evaluating integrals, both definite and indefinite!
All content in “Integration techniques”

Integration by parts

When we wanted to take the derivative of f(x)g(x) in differential calculus, we used the product rule. In this tutorial, we use the product rule to derive a powerful way to take the anti-derivative of a class of functions--integration by parts.


U-substitution is a must-have tool for any integrating arsenal (tools aren't normally put in arsenals, but that sounds better than toolkit). It is essentially the reverise chain rule. U-substitution is very useful for any integral where an expression is of the form g(f(x))f'(x)(and a few other cases). Over time, you'll be able to do these in your head without necessarily even explicitly substituting. Why the letter "u"? Well, it could have been anything, but this is the convention. I guess why not the letter "u" :)

Trigonometric substitution

We will now do another substitution technique (the other was u-substitution) where we substitute variables with trig functions. This allows us to leverage some trigonometric identities to simplify the expression into one that it is easier to take the anti-derivative of.

Division and partial fraction expansion

When you're trying to integrate a rational expression, the techniques of partial fraction expansion and algebraic long division can be *very* useful.