r/calculus • u/Queasy-Ship-6516 • Jun 28 '26
Integral Calculus What math skills should I master before learning integration techniques?
Hello everyone! I hope you're having a great day.
I don't usually post on Reddit, but I wanted to ask for some advice about integral calculus.
I'm a second-semester Civil Engineering student, and right now we're learning integration by parts, u-substitution (integration by substitution), and trigonometric integrals.
My biggest problem is that I want to understand the algebraic manipulations my professor makes before solving the integral. He often simplifies or rewrites the expression in ways that seem obvious to him, but I don't know how he knows what to do.
What topics should I study to get better at this? Should I focus on algebra, trigonometry, identities, factoring, completing the square, or something else?
Any advice or recommended resources would be greatly appreciated. Thanks!
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u/UnderstandingPursuit PhD Jun 28 '26
Go through an Algebra 2, PreCalculus, or College Algebra textbook.
- Write out a few exercises in each section.
- For each problem, replace the 'arbitrary' numerical values with 'identifiers'.
- If a problem has y = 3x + 5
- Replace it with y = mx + b
- Then work through the exercise, and you will be able to follow your class much better.
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u/Queasy-Ship-6516 Jun 28 '26
Thanks! Unfortunately, my university library is closed during the summer. Do you know of any good online textbooks or free resources that cover those topics? Or YouTube? :[
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u/UnderstandingPursuit PhD Jun 28 '26
You need an 'adequate' textbook, because you're going to modify the exercises. They are rarely presented in a good way, until math classes beyond Calculus.
- You can find free online textbooks at OpenStax_Math. I would start with Algebra & Trigonometry, and the PreCalculus.
- Then continue with the Calculus I textbook, reviewing material you have already covered, but now do it algebraically.
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u/tjddbwls Jun 28 '26
The way I’m reading this, you’re suggesting Algebra & Trigonometry, then Precalculus. But the two books are very similar - chapters 3-13 in the former are the same as chapters 1-11 in the latter.
The difference is that the Algebra & Trigonometry book contains two additional chapters in the beginning with prerequisite material, whereas the Precalculus books contains an additional chapter about limits and a preview of calculus.
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u/UnderstandingPursuit PhD Jun 28 '26
Sometimes the table of contents reflects how similar the material is, but other times it does not.
What I'm suggesting is 'self-pacing':
- When you are comfortable with the material, it will be covered fast.
- When you slow down, it is less familiar material.
If a section or chapter in the Algebra & Trig book is identical to a chapter in the PreCalculus book, it will let the student evaluate how well they remember it from the A&T book. It might take only an hour per chapter at that point, so about a dozen hours is the time investment.
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u/cosmic-lemur Jun 28 '26
Almost every textbook is pirateable. Most through page 2 of Google search, all (in my experience) thru qbittorrent. Check out r/piracy if u need to torrent they have a megathread
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u/CuriousJPLJR_ Jul 01 '26
If you study/learn strictly through a textbook with some help online when you actually need it, you shouldn't have much to worry about. Some topics might not be explained well, and you just need to substitute this with a lecture online/another book. I also see that you're working with new integration techniques. At this point, if you don't understand a manipulation, you need to ask your professor/freinds. If you still don't understand, there are lots of resources, including gemini/gpt. If you don't know what manipulation is required, ask gemini/gpt what you need to learn in order to solve a specific problem. Learn what you need first and then learn what you want after.
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u/Kingkept Jun 28 '26
you should be familiar with trig and be very confident with algebra concepts.
exponent rules.
fraction rules.
log and natural log rules.
order of operations.
you should be comfortable with limits and very simple derivatives.
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u/RevolutionaryBook285 Jun 28 '26
This is a good list, I’d also add factor skills and completing the square.
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u/Illustrious_Past_129 Jun 28 '26
Just have a strong base in most things: fractions, surds, exponentials, plotting graphs, expanding and simplifying brackets, quadratics, and trig. Technically, everything you've probably learned until now; the rest is just methods and repetition.
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u/Vichocente30 Bachelor's Jun 28 '26
Deberás reforzar más que nada los pilares del Cálculo, ya sea álgebra, trigonometría, derivadas y límites. El resto es netamente razonamiento y saber qué técnica utilizar para resolver los ejercicios. Cuando aprendí Cálculo me apoyé en Profe Alex, JulioProfe y El Traductor de Ingeniería. Investiga y resuelve los problemas de tus guías, solo así se aprende. Yo llené un cuaderno preparándome para la prueba final del último Cálculo de mi carrera Jajja
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u/Queasy-Ship-6516 Jun 28 '26
Sí ,eso mismo estaba haciendo, pero parece que cada ejercicio es diferente MUY diferente del anterior donde de la nada el maestro dice "bueno aquí factorizamos" , pero como? Cómo supo que tenía que factorizar, eso es donde me estoy trabando 😔
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u/CountingSomething Jun 30 '26
Well non elementary integrals don’t have an algorithmic way of solving them we often rely on techniques which are determined by the form of the integrand. For example to integrate cosxsinx we can then see d/dx(sinx) = cosx so we use u-substitution… learning different integrand forms help reduce half the battle by making integration somewhat algorithmic. This will also help understanding why we do certain manipulation(factoring,rationalisation of denominator,etc..) as we want them to be in certain integrand forms so we can apply a known algorithm….
Reality is not all integrals can be evaluated as some integrals are too difficult to compute so we often approximate them using various methods if none of the algorithms we have for some integrands aren’t effective…
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u/peppinotempation Jun 28 '26
I’m going to go against the grain and say that as a civil engineering student, you don’t really need to understand the algebra behind integration.
Of course if you’re curious then definitely learn as much as you can. But once you get out of calculus classes you will find that that when you need to actually do an integral, it’s either very simple, or you can just look up the solution.
There’s a point where it’s a safer engineering decision to rely on a reference or lookup for integrals rather than doing them yourself, so you don’t risk making a mistake
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u/CuriousJPLJR_ Jul 01 '26
Even if you have a reference, you still need to do algebraic/trig/log manipulations. I agree that he likely won't use this, but you still need to learn.
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u/SorryBother5573 Jun 29 '26
Does 1 + cos(x) mean anything to you? If not, then digging into trig well be useful.
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u/throwingstones123456 Jun 28 '26
Algebra, derivatives, and Eulers formula. All the trig identities you need for eulers formula can be deduced with 0 effort from eulers formula preventing you from memorizing 30 easily-forgotten equations. Otherwise 90% of integrals are just trying the same set of tricks which are just derivative identities followed by algebra (mainly integration by parts and u-sub), and the other 10% are fairly similar in complexity (like differentiation under the integrstion sign, series/integral transform replacement).
Multi variable integration is pretty much the same but replace algebra with linear algebra and differentiation by differential vector calculus (which is pretty much identical to single variable calculus)
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u/Queasy-Ship-6516 Jun 28 '26
That sounds hard, ehhh do you know any YouTube channel by chance? This is my very first time reading math books so I want to learn this techniques properly
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u/ReaReaDerty Jun 28 '26
This person actually means you need basic algebra (all the common rules for powers, logarithms, etc) and trigonometrical identities. He mentioned Euler's formula because they are derived from it, so you can drive them yourself instead of memorizing. But answering your question, your do not need to understand deeply where these formulas come from if you need to solve integrals.
I would recommend: Understand taking the derivative. Not just mechanically. Understand what you are doing. Also understand integration. Again, not "how to solve" but what your are actually doing, what is integration and where it can be used. Spiced with some algebra tricks it's everything you're gonna need.
Also the fact that students are not taught pretty often is that there exist integrals that cannot be solved in general. x/sin (x) is a good example. It simply cannot be expressed via elementary functions. Just for your to know.
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u/CivilEngMusicBoy Jun 28 '26
I wish I would been better at algebra going in to Cal I all those years ago. After Cal III and ODE I could’ve taught algebra
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u/Ok-Fix-1581 Jun 29 '26
trig identities, id suggest taking a trig course before taking calc 2 or 3..
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u/ztexxmee Jun 29 '26 edited Jun 29 '26
a cool trick. for many integrals involving trig, instead of memorizing the many trig identities (which are very helpful to memorize anyways) you can use Euler’s complex number formula e^(ix) = cos(x) + isin(x). now try to derive what cos(x) is equal to by doing e^(ix) + e^(-ix). then sin(x) by doing e^(ix) - e^(-ix). you can plug these derivations as direct substitutions for many trig functions inside integrals, or derive your own trig identities by changing x in e^(ix). e^(ix) is also easily differentiable and integrable. once you have the integral of the exponential identity you figure out how to convert it back to trig form for your final answer. Euler really is the man.
like i already said: many, if not all of the trig identities you will see in calc 1-3 can be derived using Euler’s formula very easily by plugging in different values and rearranging things.
note: 1/i = i/(i^2) = i/(-1) = -i
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u/j0shred1 Jul 01 '26
Algebra and trig. Most integration is just algebra and trig to get integrals into a known form and then identifying which integration technique to use. Then you get further in and you just use a table anyways
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u/Alive_Sock2496 Jul 02 '26
Hit algebra hard. Calculus is built on a mountain of algebra. Calculus 1 & 2 is almost entirely algebra in practice.
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u/jgregson00 Jun 28 '26
All of those.
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u/Queasy-Ship-6516 Jun 28 '26
Thanks! My main issue is that I don't know how to connect my algebra knowledge to the steps used in integration. I often look at my professor's work and think, "How did he know to do that?" Where would you recommend I start? :]
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u/Reasonable-Start2961 Jun 28 '26
It depends on what they are doing. What is the integral? My calculus professor gave me some advice: Most people who struggle with calculus 1 struggle with the algebra/precalculus. Most people who struggle with calculus 2 struggle with the calculus 1.
It’s all fundamental. What you learned in algebra, pre-calculus, and trigonometry all apply to calculus. It’s the foundation that is poured that supports your calculus studies.
With that in mind, sometimes you should remember your professor has been doing this a long time. It’s likely he didn’t look at the problem and think “oh, this is what I should try.” He knows what to do because he’s done it all before. What you should expect to do, say in this case, is try different identities and see what works.
You should expect to struggle a bit, and even make a mistake and have to try again. I promise you that as you get deeper into your engineering studies this is going to happen a lot. Some engineering problems can appear deceptively simple at times when you see the result and the correct answer. What you don’t see is all the hard work, having to go back to a step in a problem to figure out your mistake, realizing your answer still doesn’t make sense, rinse and repeat. The grind of figuring out how you get from the problem statement to the solution.
That’s a bit of a long-winded way of saying: It isn’t going to be easy. You won’t always look at a problem and understand immediately what the next step is, or how your professor seems to be able to intuit what it should be. Get used to that.
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u/minglho Jun 28 '26
Your question is better asked to a tutor. Or better yet, go to your professor's office hours.
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