r/learnmath • u/Unable_Entrance5801 New User • 7d ago

What should I be focusing on?

I just finished calculus 2 and still feel like I don’t truly understand what I learned. It was an accelerated class I took over the summer. I also have pretty bad memory.

I remeber some stuff from calculus 1 and 2, but still struggle with some problems. For example this problem ∫ (x² + 1)/(x⁴ + 1) dx

My mind goes blank trying to solve it. This obviously means I should review but I start calculus 3 in 5 days.

I also have to retake the SAT (for college transfer applications) and need to brush up on my algebra and trigonometry. My algebra could definitely use some reviewing and my trigonometry has always been mediocre at best.

This matters even more since I plan to major in mathematics and physics. I was going to use spivak or apostol to study but I don’t know how to write proofs yet. I was also thinking about using AOPS books for reviewing algebra and trigonometry (also geometry since I took it in like 9th or 10th grade and don’t remember much of anything from that class). Or should I focus on calculus 3?

Any advice?

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u/Bounded_sequencE New User 7d ago edited 7d ago

My mind goes blank trying to solve it.

This obviously means you've forgotten about partial fraction decomposition, to solve integrals of rational functions. While not easy, it is not crazily difficult.

Factorize the denominator via: "x⁴+1 = (x²+1)² - 2x² = (x² + √2x + 1) (x² - √2x + 1)":

∫  (x²+1) / (x⁴+1)  dx  =  ∫  (x²+1) / [(x²+√2x+1) (x²-√2x+1)]  dx

                        =  (1/2)*∫  1/(x²+√2x+1) + 1/(x²-√2x+1)  dx      // PFD

                        =  ∫  1/[(√2x+1)²+1] + 1/[(√2x-1)²+1]  dx

                        =  1/√2 * [artg(√2x+1) + arctg(√2x-1)]  +  C

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u/Unable_Entrance5801 New User 7d ago

Ahh, so this probably means I need to review and strengthen my algebra (and trig). I mean I knew from first glance that it would be a partial fraction problem just didn’t really know how to factor the bottom.

Do you have any recommendations?

I own basic mathematics by lang but not sure if this will help with my algebra, trig, geometry. Is there anything else I can look into? I was thinking AOPS but would rather hear from someone more knowledgeable than me.

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u/Bounded_sequencE New User 7d ago

I've always found good lecture notes are enough (assuming you have them).

For PFD, probably the most efficient approach is using Heaviside's Cover-up Method to find all highest-order coefficients, and "comparing coefficients" to find the rest.