Calculus 2 is where many university students hit a wall. The integration techniques are complex. The series and sequences unit is unlike anything before it.
At Fit Minds Academy our tutors help university students across Canada get through Calculus 2 — whether you’re in MAT136, MAT237, MATH 1XX3, or any equivalent second year calculus course.
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Calculus 2 is the second university calculus course. It picks up where left off — after derivatives and basic integration — and goes much deeper into integration techniques, infinite series, and applications.
University of Toronto (standard)
U of T (advanced stream)
McMaster University
Western University
Calculus 1 vs Calculus 2: Calculus 1 focuses on derivatives and an introduction to integration. Calculus 2 goes much deeper into integration — new techniques, applications, and theory. It also introduces infinite series which is an entirely new area of mathematics that most students find the hardest part of the course.
We offer tutoring for all university mathematics courses across Canada.
Integration by parts, trig integrals, trig substitution, partial fractions
Convergence and divergence of integrals with infinite limits
Area between curves, volumes of revolution, surface area of revolution, arc length
Convergence tests, power series, Taylor series, Maclaurin series, radius of convergence
Parametric equations, polar coordinates, area and arc length in polar form
Separable differential equations, slope fields, applications
This is the first major hurdle in Calculus 2. You learned basic integration in Calculus 1. Now you need a toolkit of techniques for integrals that cannot be solved with simple rules.
Integration by parts is the integration equivalent of the product rule.
∫u dv = uv − ∫v du
L
Logarithmic functions — ln(x)
I
Inverse trig functions — arctan(x), arcsin(x)
A
Algebraic functions — polynomials
T
Trigonometric functions —sin(x), cos(x)
E
Exponential functions — eˣ
Choose u as the first type that appears in the integrand. Everything else is dv.
∫x eˣ dx
u = x → du = dx
dv = eˣ dx → v = eˣ
= xeˣ − ∫eˣ dx = xeˣ − eˣ + C = eˣ(x − 1) + C
Choosing u and dv backwards. If you choose u as the exponential and dv as the polynomial, the integral on the right becomes more complicated — not simpler. LIATE prevents this.
Integration by parts is the integration equivalent of the product rule.
Use trig substitution to handle integrals containing √(a²−x²), √(a²+x²), or √(x²−a²).
1 − sin²θ = cos²θ
1 + tan²θ = sec²θ
sec²θ − 1 = tan²θ
After substituting — integrate in terms of θ, then convert back to x using a reference triangle.a
Partial fractions decomposes a rational function into simpler fractions that are easy to integrate.
The integrand is a fraction where the degree of the numerator is less than the degree of the denominator, and the denominator can be factored.
1.
Factor the denominator completely
2.
Write the partial fraction decomposition — one term per factor
3.
Multiply through and match coefficients to find constants
4.
Integrate each simple fraction separately
∫x eˣ dx
Factor denominator:
(x+1)(x+2)
Decompose:
(3x+5)/((x+1)(x+2)) = A/(x+1) + B/(x+2)
Solve:
A = 2, B = 1
∫ = 2ln|x+1| + ln|x+2| + C
An improper integral has either an infinite limit of integration or an integrand that becomes infinite within the interval.
Replace the infinite limit with a variable t, evaluate the definite integral, then take the limit as t → ∞.
∫₁^∞ 1/x² dx = lim(t→∞) ∫₁ᵗ x⁻² dx = lim(t→∞) [−1/x]₁ᵗ
= lim(t→∞) (−1/t + 1) = 1
This integral converges to 1.
If the limit exists and is finite — the integral converges.
If the limit is infinite or does not exist — the integral diverges.
p > 1
p ≤ 1
Use when the region is rotated around an axis and there is no gap between the region and the axis.
V = π ∫ₐᵇ [f(x)]² dx
(rotating around x-axis)
Use when there is a gap — an inner radius and an outer radius.
Use when rotating around a vertical axis and integrating with respect to x is more convenient.
S = 2π ∫ₐᵇ f(x) √(1 + [f'(x)]²) dx
This formula consistently surprises students with how involved the algebra gets. The derivative term inside the square root often requires trig substitution or integration by parts to evaluate. It is one of the most calculation-intensive topics in the course.
L = ∫ₐᵇ √(1 + [f'(x)]²) dx
Notice the arc length formula is the same as surface area of revolution but without the 2πf(x) factor.
A separable differential equation can be split so that all y terms are on one side and all x terms are on the other.
1
Separate variables — get all y terms with dy on one side, all x terms with dx on the other
2
Integrate both sides
3
Solve for y if possible
4
Apply initial conditions if given to find the constant C
dy/dx = xy
Separate:
dy/y = x dx
Integrate:
ln|y| = x²/2 + C
Solve
y = Ae^(x²/2) where A = ±eᶜ
Exponential growth and decay. Population models, radioactive decay, and cooling problems all use the same form dy/dt = ky, which gives y = y₀eᵏᵗ.
This is the unit most Calculus 2 students find hardest. It is conceptually different from everything before it and requires a new way of thinking.
A sequence is an ordered list of numbers generated by a formula. The key question is always whether the sequence converges (approaches a finite limit) or diverges.
lim(n→∞) aₙ = L
means the sequence converges to L
A series is the sum of the terms of a sequence. S = Σaₙ from n=1 to ∞. The key question is the same — does the sum converge to a finite value or grow without bound?
S = Σaₙ
When to use: Always check first
If lim aₙ ≠ 0, series diverges. If = 0, inconclusive.
When to use: aₙ = f(n) where f is positive, continuous, decreasing
Converges iff ∫f(x)dx converges
When to use: Σ 1/nᵖ
Converges if p > 1, diverges if p ≤ 1
When to use: Terms comparable to known series
Compare to geometric or p-series
When to use: Rational or algebraic terms
lim(aₙ/bₙ) = L > 0 means both behave the same
When to use: Factorials or exponentials
L < 1 converges, L > 1 diverges, L = 1 inconclusive
When to use: Series alternates sign
Converges if terms decrease to zero
When to use: aₙ raised to a power
L < 1 converges, L > 1 diverges
1.
First check the divergence test — if terms don’t go to zero, done
2.
Look at the form — factorial/exponential → ratio test. Power of n → root test or p-series
3.
Alternating? → alternating series test
A power series is an infinite series of the form Σcₙ(x−a)ⁿ. It converges for some values of x and diverges for others.
The radius of convergence R is the distance from the center a within which the series converges. Use the ratio test to find R.
R = lim(n→∞) |cₙ/cₙ₊₁|
The interval of convergence is (a−R, a+R)
Important: You must check the endpoints separately by substituting them directly into the series.
A Taylor series represents a function as an infinite polynomial centered at x = a.
f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ
A Maclaurin series is a Taylor series centered at a = 0.
eˣ
1 + x + x²/2! + x³/3! + …
sin(x)
x − x³/3! + x⁵/5! − …
cos(x)
1 − x²/2! + x⁴/4! − …
1/(1−x)
1 + x + x² + x³ + …
(geometric)
ln(1+x)
x − x²/2 + x³/3 − …
Calculus 2 cheat sheet series section — the most requested part of any formula sheet.
This is one of the most searched questions by students entering second year.
Core focus
Derivatives and basic integration
Hardest topic
Related rates and implicit differentiation
Computation level
High
Conceptual difficulty
Moderate
Most students say
Hard but learnable
Core focus
Advanced integration and infinite series
Hardest topic
Sequences and series convergence tests
Computation level
Very high
Conceptual difficulty
High — especially series
Most students say
Harder than Calculus 1
Most university students say yes. The integration techniques require a larger toolkit and more judgment about which method to apply. The series and sequences unit introduces genuinely new mathematical thinking. And the volume and surface area of revolution problems combine multiple skills at once under time pressure.
Yes — Calculus 2 is one of the hardest courses in the first two years of university. Here’s what trips students up most:
Failure rates in Calculus 2 are higher than Calculus 1 at most universities. The series unit is where most students lose their grade. Starting convergence test practice early — not just before the exam — is the single most effective strategy for passing.
This is searched constantly by struggling students. The answer — do not write down what the professor writes. Write down the method.
For every example, write a one-line summary of why that technique was chosen. That is the note that helps you on an exam. Formulas are on the formula sheet. The judgment of when to use them is what you need to capture.
📊 Real Result: One of our students came to us after failing her MAT136 midterm with a 41%. She had memorized all the integration techniques but froze on exams because she couldn’t decide which one to use. After 3 sessions building a personal decision flowchart for integration she scored 78% on her Calculus 2 final. The technique wasn’t the problem — the strategy was.
Here are the exact resources our students use every semester — all completely free.
Every formula from the full Calculus 2 course organized by topic. This is the most complete calculus 2 formula sheet pdf available for Ontario university students.
What’s inside:
A complete Calculus 2 practice exam with full step-by-step solutions. Covers all major topics at easy, medium, and hard difficulty.
What’s inside:
Not sure where to start your Calculus 2 final exam review or Calculus 2 midterm review? This checklist covers every testable topic.
What’s inside:
Clear Calculus 2 notes written in plain English. Every concept explained simply with worked examples throughout.
What’s inside:
Start with the exam review checklist to find your weakest topics. Use the notes to review those topics — paying attention to when each technique applies, not just how. Check the calculus 2 cheat sheet to make sure every formula is familiar. Then work through the practice exam under timed conditions with no notes. That's exactly how our students prepare — and it works.
These free resources are a great start. But nothing replaces a tutor who works through it with you live — using your actual course content and your professor’s exam style.
No hidden fees. No long contracts. Just results.
💯 100% money-back guarantee on your first lesson
We offer in-person Calculus 2 tutoring across Mississauga, Toronto, Brampton, Oakville, Richmond Hill, Scarborough, North York, and Burlington. For students in Hamilton, Markham, Newmarket, Guelph, Waterloo, Calgary, Edmonton, Ottawa, Montreal, Winnipeg, and Vancouver — fully interactive online sessions are available. Wherever you are in Ontario or Canada, we’re here.
Calculus 2 covers six main areas — advanced integration techniques (by parts, trig substitution, partial fractions), improper integrals, applications of integration (volume and surface area), sequences and series with convergence tests, power and Taylor series, and an introduction to differential equations.
Yes — most university students find Calculus 2 harder than Calculus 1. The integration techniques require judgment about which method to apply. The series unit is conceptually different from all prior math. And the volume of revolution problems require three-dimensional visualization. With consistent practice and a tutor who builds your problem-selection instincts, most students improve significantly.
Most students say Calculus 2 is harder. is computationally intensive but the methods are straightforward. Calculus 2 requires more judgment — especially in choosing the right integration technique and the right convergence test. The series unit introduces entirely new mathematical thinking that has no direct parallel in earlier courses.
The radius of convergence R describes how far from the center of a power series the series converges. Use the ratio test — R = lim|cₙ/cₙ₊₁| — to find it. The series converges for all x within distance R from the center. Always check the endpoints separately by substituting them directly.
Integration by parts is the integration version of the product rule. Formula: ∫u dv = uv − ∫v du. Use the LIATE rule to choose u — logarithmic, inverse trig, algebraic, trigonometric, exponential in that priority order. It is used when the integrand is a product of two different types of functions.
Separable differential equations can be split so all y terms go to one side and all x terms go to the other. Separate, integrate both sides, and solve for y. They are the entry point into differential equations and appear on most Calculus 2 final exams.
The disk method is used when rotating a region with no gap between the region and the axis of rotation. The washer method is used when there is a gap — creating a hole in the solid. The washer method subtracts the inner radius squared from the outer radius squared before integrating.
A Taylor series represents a function as an infinite sum of polynomial terms derived from the function’s derivatives at a single point. A Maclaurin series is a Taylor series centered at zero. The most important ones to memorize are eˣ, sin(x), cos(x), and 1/(1−x).
Yes. We offer Calculus 2 tutoring online for university students anywhere in Canada. Sessions are fully interactive with a shared digital whiteboard so every step is worked out live in real time.
Yes. Our tutors help with Calculus 2 assignments, midterm prep, MAT136 and MAT237 exam review, and Calculus 2 final exam preparation. We work through every problem step by step so you understand the solution — not just copy it.
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Whether you need help with integration by parts, the ratio test, radius of convergence, surface area of revolution, separable differential equations, or the whole course — we’re here.
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