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manpreet Best Answer 2 years ago

I've learnt the following in terms of integration;

What are integrals, and what do they represent?
Indefinite integrals as the opposite of derivatives
Using the power rule for derivatives to provide the power rule for integration
Integral as infinite sum
Area under curve, and area between two curves
Volume of solid of revolution
Integration by substitution
Integration by parts
Integration by partial fraction expansion
Improper integrals

I'm wondering where to go from here. I know the basics of numerical methods, and the fundamental theorem as well. I'm looking to go further in terms of analytical integration, but I don't know which topic would be next. Any ideas?

EDIT: I've checked the syllabus for AB/AP calculus, but it only lists very vague ideas of the concepts, and not sub-topics.

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manpreet 2 years ago

Maybe it would be nice to see some stuff with complex numbers, after line integrals. For example, we can simplify some calculations, using $e^{i x} = \cos x + i \sin x$ . How can you calculate $\int e^{x} \cos x d x$ without integration by parts? Is there an easy way to find an expression for the $n$ -th derivative of $e^{x} \sin x$ ? What means to differentiate a function in $C$ ? Can we derivate something like $x^{2} y + i \frac{y}{x}$ ? And so on. I hope this motivates you a bit (:

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Learning integration, and have gotten to a certain point. What is the next logical step?

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manpreet Best Answer 2 years ago

manpreet 2 years ago

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