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

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.

• I would suggest multi-variable calculus and multi-variable integration. Then, you could go for a more abstract analysis: measure theory. – Alessandro Flati Apr 9 '14 at 10:27
• Try to read up on line integrals and alike, or dwell deeper into the theory you already know =) folk.ntnu.no/oistes/Diverse/Integral%20Kokeboken.pdf Part II here is a perhaps poor attempt to explore the basics deeper.You probably need to use google translate a few places. "Handbook of integration" is another great book or "Vector Calculus" if that's the path you want to take. – N3buchadnezzar Apr 9 '14 at 17:42
• @N3buchadnezzar - I see your cookbook is getting awesomer. You really, REALLY should make an English copy, you know. Stuff like this is invaluable. That said; yes. I will definitely use it for reference. I'm ashamed that I'd forgotten about it, actually. – Alec Apr 9 '14 at 17:50

Maybe it would be nice to see some stuff with complex numbers, after line integrals. For example, we can simplify some calculations, using $e^{ix} = \cos x + i \sin x$. How can you calculate $\int e^x \cos x \mathrm{d}x$ 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 $\mathbb{C}$? Can we derivate something like $x^2y + i\frac{y}{x}$? And so on. I hope this motivates you a bit (: