The Calculus II survival guide: integrals and series without the tears

A student came to me convinced she was "bad at Calc II." She'd failed the first midterm, the integration one, and had quietly decided she wasn't a math person. So I put an integral in front of her — a rational function, the kind partial fractions eats for breakfast — and watched. She knew partial fractions. She knew integration by parts. She knew u-substitution cold. She just had no idea which one to reach for, so she froze, tried parts because it was freshest in her memory, got tangled, and ran out of time. She wasn't bad at Calc II. She was missing a decision procedure.

That's the thing nobody warns you about. Calculus II has a reputation as the great weed-out course, the one that ends a lot of engineering and pre-med ambitions. But here's what I've learned teaching it for over a decade: for most students, Calc II isn't hard because the individual techniques are hard. It's hard because it's the first math course that's mostly about decision-making. Calc I gave you one tool at a time and told you when to use it. Calc II hands you a whole toolbox and a problem that doesn't say which drawer to open. Master the choosing, and the course shrinks back to a manageable size. This guide is about the choosing.

The integration decision tree

When you see an integral on an exam, do not start computing. Spend ten seconds classifying it first. The standard strategy — laid out clearly in references like Paul Dawkins' widely-used Lamar notes — is roughly an ordered checklist. Run down it in order and stop at the first technique that fits.

Step 0 — Simplify and look for a standard form. Before anything clever, can you simplify the integrand with algebra or a trig identity? Is it already a form you have memorised (a power of x, sin, cos, e^x, 1/x)? A shocking number of "hard" integrals are an algebra step away from trivial. Multiply out, split a fraction, apply sin²+cos²=1 — then re-examine.

Step 1 — Try u-substitution. This is your first real move, and it's the most common technique in the course, so look for it before anything more elaborate. The tell: is there a function inside the integrand whose derivative is also sitting there (up to a constant)? If you see something like x·e^(x²) — the x out front is, conveniently, the derivative of the x² inside — that's a u-sub screaming at you. Set u = the inner function, and if du matches what's left, you're done. Always check for u-sub before reaching for the heavier machinery.

Step 2 — Is it a product begging for integration by parts? If u-sub doesn't fit and the integrand is a product of two different kinds of functions — a polynomial times a trig function, a polynomial times an exponential, or anything times a logarithm or inverse-trig function — reach for integration by parts, ∫u dv = uv − ∫v du. The classic signals: x·sin(x), x²·e^x, ln(x) (parts with dv = dx), arctan(x). A useful mnemonic for picking which factor is "u" is LIATE — Logarithmic, Inverse-trig, Algebraic, Trig, Exponential — pick u from whichever comes first in that list, and let the rest be dv.

Step 3 — Does it have a telltale square root? Trig substitution. If you see a square root of the form √(a² − x²), √(a² + x²), or √(x² − a²), that's the fingerprint of trig substitution. Each shape maps to a specific substitution (x = a·sinθ, a·tanθ, a·secθ respectively) that turns the root into a clean trig expression via a Pythagorean identity. The moment you spot one of those three radical shapes and no simpler u-sub works, trig sub is your tool.

Step 4 — Is it a rational function? Partial fractions. If the integrand is a polynomial divided by a polynomial (a rational function) and you can't u-sub your way out, partial fractions breaks it into a sum of simpler pieces you can integrate. First check the degree: if the top's degree is ≥ the bottom's, do polynomial long division first, then decompose the remainder. Factor the denominator, split into partial fractions, integrate term by term.

Step 5 — Combine and retry. Real integrals are sometimes a u-sub into an integration-by-parts, or a trig sub that leaves you with partial fractions. If your first technique stalls partway, you're often one substitution away from a form an earlier step handles. Cycle back through the list with the new expression.

Here's the whole thing as a table you can pin above your desk:

The reason this checklist works is psychological as much as mathematical: it gives you something to do when you freeze. Instead of staring at the integral hoping inspiration strikes, you run the list. Most exam integrals reveal themselves by step 2. This decision-first habit is the single biggest thing I drill in one-on-one calculus sessions — the techniques themselves are in the textbook; the triage is what's missing.

The series-convergence test picker

The second half of Calc II — infinite series — has the exact same problem in a new costume. There are something like ten convergence tests, and the panic comes not from any single test but from not knowing which test fits which series. So, same fix: a picker keyed to the shape of the series.

Look at the general term aₙ and ask, in roughly this order:

First — do the terms even go to zero? Apply the Divergence Test. If aₙ does not approach 0, the series diverges, full stop — you're done. (Warning, and exams love this trap: if aₙ does go to 0, this test tells you nothing. It can prove divergence but never convergence.)

Is it a recognisable named series? Two you should spot instantly:

• Geometric series (each term is a constant ratio r times the previous): converges if |r| < 1, diverges otherwise. Clean and decisive.
• p-series (terms of the form 1/n^p): converges when p > 1, diverges when p ≤ 1. The boundary case p = 1 is the harmonic series 1 + 1/2 + 1/3 + …, which diverges — a fact worth burning into memory, because the terms shrink to zero yet the sum still runs off to infinity.

Does it have factorials or terms raised to the n-th power? That's the signature of the Ratio Test (look at the limit of |aₙ₊₁/aₙ|). Factorials telescope beautifully under a ratio, so anything with n! anywhere is a ratio-test candidate. If the entire term is something raised to the n-th power, like (…)ⁿ, the Root Test is often even cleaner — and it's strictly stronger than the ratio test, meaning whenever the ratio test gives an answer the root test does too.

Does it look like a p-series or geometric series but messier? Use a Comparison Test or Limit Comparison Test — bound it against, or compare its growth to, a series you already understand. Limit comparison is the gentler one: form the limit of aₙ/bₙ against a known series bₙ and read off the verdict.

Can you integrate the term as a function? If aₙ = f(n) for a function that's positive and decreasing, the Integral Test ties the series' fate to an improper integral you can actually evaluate. Great for things like 1/(n·ln n) that the other tests fumble.

Are the signs alternating? For a series whose terms flip sign each time, the Alternating Series Test applies: if the absolute values decrease monotonically to zero, it converges. And keep the distinction straight — absolute convergence (the series of absolute values converges) is stronger than conditional convergence (it converges, but only thanks to the alternating signs).

Same lesson as integration: you're not memorising ten tests to apply blindly — you're learning to read the shape of the series and pick the one test built for it. Run the picker top to bottom and stop at the first match.

Two Taylor-series intuitions that make the notation click

Taylor series is where a lot of students decide the notation has personally wronged them. Two intuitions do most of the un-confusing.

One: a Taylor series is just a polynomial that mimics a function near a point. That's the whole idea. You can't easily compute e^(0.1) by hand, but you can compute a polynomial. So you build a polynomial that matches the function's value, then its slope, then its curvature, then the next derivative, and so on at a chosen point. Each new term sharpens the mimicry. The intimidating Σ formula is bookkeeping for "match the k-th derivative at the centre" — nothing deeper. Match more derivatives, hug the curve more closely.

Two: the more terms you keep, the wider the region where the polynomial is a good fake — and the interval of convergence tells you how far you can trust it. A few terms match the function only very near the centre; add terms and the approximation stays faithful further out. The interval of convergence is precisely the range of x where the infinite series actually equals the function instead of drifting off. So when a problem asks for an interval of convergence, it's really asking "over what range is this polynomial impersonation trustworthy?" That reframing turns an abstract exercise into a concrete question, and suddenly the radius-of-convergence computation has an obvious point.

The office-hours secret

Here's the thing I most want you to take from this, and it's the one that surprises students every single time. When a student is hopelessly lost in Calc II, the gap is almost never in Calc II. It's a single crack in the foundation underneath — an algebra skill or a Calc I idea that everyone assumed was solid and nobody checked.

The student who can't finish partial fractions usually can't factor the denominator fluently — that's an Algebra II skill, not a Calc II one. The student whose integration by parts always collapses often has a shaky grip on the derivative and integral rules from Calc I that parts depends on. The student who can't evaluate the improper integral in the Integral Test is fine on the test itself — they just never got comfortable with limits. The Calc II machinery sitting on top is working perfectly; it's the one missing brick below that brings the wall down.

This is why "I'm just bad at Calc II" is almost always the wrong diagnosis. You're typically one specific, findable gap away from things clicking — and once you find and patch it, a whole cluster of problems opens up at once, because that one gap was quietly sabotaging five different topics. The fastest tutoring sessions I run are the ones where we go looking for that brick instead of re-teaching the technique that isn't actually the problem.

Exam strategy: triage by recognisable technique

A concrete tactic for the test itself, because a good decision procedure is worthless if you spend all your time on question 3. Do a fast triage pass before you compute anything. Read every problem, and next to each one jot the technique you'd use — "u-sub," "parts," "ratio test," "no idea yet." Then solve in order of confidence: the ones you instantly recognise first, banking guaranteed marks, and save the genuine head-scratchers for last.

This does two things. It guarantees you capture the points you've already earned in your head, and it lets the back of your mind chew on the hard ones while your hands are busy on the easy ones — you'll be amazed how often the technique for question 6 surfaces while you're calmly working question 2. Freezing on the first hard problem and burning twenty minutes there is how good students walk out having left a third of the exam blank. Recognise, triage, bank the sure points, then circle back.

The recap: Calc II is a decision-making course wearing a difficulty costume. For integrals, run the checklist — simplify, u-sub, parts, trig sub, partial fractions, combine. For series, read the shape and pick the matching test, starting with whether the terms even go to zero. Lean on the two Taylor intuitions: it's a polynomial impersonator, and the interval of convergence is how far you can trust it. And if you feel hopelessly lost, suspect one foundational gap rather than a global failing — because that's almost always what it is.

If you've decided you're "not a math person" on the strength of one bad midterm, I'd push back on that gently — in my experience it's nearly always a single fixable gap, not a verdict. Finding it is genuinely faster with someone watching you work a problem than alone with a textbook, and that's exactly what I do in one-on-one calculus tutoring. The first session is free, so it costs you nothing to find out which brick is missing.