Course · Electrical engineering track

Calculus II for Engineers

Integration techniques, applications, infinite series, parametric and polar curves, and a first pass at differential equations — with every unit pointed at the place an electrical engineer actually meets it. Partial fractions exist so you can invert a Laplace transform. Series exist so you can linearize a diode. Polar coordinates exist so you can read a Nyquist plot.

Units

Unit 1

Techniques of Integration

Parts, trig integrals, trig substitution, partial fractions, improper integrals, numerical quadrature.

Unit 2

Applications of Integration

Area, volume, arc length, work, centroids — and RMS, average power, charge and stored energy.

Unit 3

Sequences & Infinite Series

Convergence tests end to end, geometric series, and the strategy for choosing a test under exam pressure.

Unit 4

Power Series & Taylor Expansions

Radius of convergence, the seven series worth memorizing, error bounds, linearization, and Euler’s formula.

Unit 5

Parametric, Polar & Complex

Parametric calculus, polar area and arc length, complex arithmetic, De Moivre, roots of unity, phasors.

Unit 6

Intro to Differential Equations

Separable, first-order linear, integrating factors, second-order constant-coefficient — solved as RC, RL and RLC circuits.

Practice exams

One per unit, ten questions each, ~90 minutes, full worked solutions hidden behind a click so you can actually test yourself. Roughly two thirds pure technique and one third applied to circuits — close to the balance a real engineering midterm uses.

Exam 1

Integration techniques

10 questions · 90 min

Exam 2

Applications

10 questions · 90 min

Exam 3

Sequences & series

10 questions · 90 min

Exam 4

Power & Taylor series

10 questions · 90 min

Exam 5

Parametric, polar, complex

10 questions · 90 min

Exam 6

Differential equations

10 questions · 90 min

What you need coming in

Calculus I, and specifically these six things. If any is shaky, patch it now — every one of them shows up in week one.

The derivative rules cold, especially chain rule and implicit differentiation. Integration is pattern-matching against derivatives you already know.
The Fundamental Theorem of Calculus, both parts, and what each one is for.
u-substitution — this is the one technique from Calc I you will use in literally every unit.
Limits, including L’Hôpital’s rule. Half of the series unit is limits in disguise.
Trig identities: Pythagorean, double-angle, half-angle. Unit 1 leans on them hard.
Algebra with exponentials and logs. e^a+b = e^ae^b is not optional in EE.

Core antiderivative table

These are the ones you should recognize instantly, without deriving. Everything in Unit 1 is machinery for converting an unfamiliar integral into one of these. (Constant of integration omitted throughout.)

Integral	Result	Integral	Result
∫ `x`ⁿ d`x`	`x`ⁿ⁺¹`n`+1, `n` ≠ −1	∫ 1`x` d`x`	ln\|`x`\|
∫ `e`^ax d`x`	1`ae`^ax	∫ `a`^x d`x`	`a`^x / ln `a`
∫ sin `x` d`x`	−cos `x`	∫ cos `x` d`x`	sin `x`
∫ sec² `x` d`x`	tan `x`	∫ csc² `x` d`x`	−cot `x`
∫ sec `x` tan `x` d`x`	sec `x`	∫ csc `x` cot `x` d`x`	−csc `x`
∫ tan `x` d`x`	ln\|sec `x`\|	∫ cot `x` d`x`	ln\|sin `x`\|
∫ sec `x` d`x`	ln\|sec `x` + tan `x`\|	∫ csc `x` d`x`	−ln\|csc `x` + cot `x`\|
∫ d`xa`² + `x`²	1`a` arctan(`xa`)	∫ d`x`√(`a`² − `x`²)	arcsin(`xa`)
∫ sinh `x` d`x`	cosh `x`	∫ cosh `x` d`x`	sinh `x`

The three that get memorized wrong

∫ sec x dx — the answer is a logarithm, not a power of sec. Nobody derives this mid-exam; memorize it.
∫ dx/(a²+x²) has a 1/a out front; ∫ dx/√(a²−x²) does not. Losing that factor is the single most common arithmetic slip in Unit 1.
∫ x⁻¹ dx is the exception to the power rule. Applying xⁿ⁺¹/(n+1) here gives division by zero.

Why this course matters for EE specifically

Calculus II is the last purely mathematical course before the tools become circuit theory. Here is the map, so you know what you are buying with each unit:

Unit	Where it lands in the EE curriculum
1 — Integration techniques	Partial fractions is inverse Laplace transform. You will do it hundreds of times in Signals & Systems and Control Theory. Integration by parts on `e`^atcos `bt` is transient circuit response.
2 — Applications	RMS voltage and average power. Every AC power calculation, every spec sheet, every meter reading. Also charge as ∫`i` d`t` and energy as ∫`p` d`t`.
3 — Series convergence	Geometric series is feedback and the z-transform. Closed-loop gain, infinite ladder networks, whether a discrete filter is stable.
4 — Taylor series	Small-signal analysis. Every transistor and diode model you will ever use is a first-order Taylor expansion about an operating point. Plus Euler’s formula, which is the whole basis of phasors.
5 — Parametric, polar, complex	Impedance lives in the complex plane. Nyquist and polar plots, antenna radiation patterns, Lissajous figures on a scope, roots of unity behind the DFT.
6 — Differential equations	Every circuit with a capacitor or inductor is an ODE. RC time constants, RL current build-up, and the overdamped/critically damped/underdamped trichotomy of RLC.

How to work through this

Read the unit page, then close it and do the practice exam cold — no notes, timed. Grade honestly, and for every miss, go back to the section it came from rather than just reading the solution. The exams are written so that each question maps to exactly one section of the unit page, which makes that loop fast.