Techniques of Integration
Develop a reliable diagnostic instinct for choosing and executing integration methods. Move past basic calculus drills to tackle rational forms, trigonometric substitutions, and multi-step analytic challenges without getting stuck.
Like Duolingo, but for Techniques of Integration. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
24 levels across 3 sections, about 48 minutes end to end, roughly 10 days at five minutes a day. It moves through The Workhorse Methods in Action, Algebraic and Geometric Transforms, and Strategic Diagnosis and Harder Problems. It assumes you already know the basics.
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Key ideas in Techniques of Integration
- A missing numerical constant can be balanced by pulling its reciprocal outside the integral.
- Solving du = 2x dx gives x dx = (1/2) du, replacing the remaining x terms cleanly.
- Variable terms must never linger outside the integral after transforming to u.
- Invertible substitution equations allow writing x explicitly as a function of u.
- Replacing leftover x factors with (u - c) turns products into easily integrated polynomials.
- Distributing u^(1/2) across (u - 1) converts the integrand into standard power rule terms.
- Completing the square on an unrooted quadratic creates the form 1 / (u^2 + a^2), which integrates to arctangent.
- Completing the square under a radical creates 1 / sqrt(a^2 - u^2), integrating to arcsine.
- Numerators matching the quadratic derivative require logarithmic integration instead of completing the square.
- Whenever an integrand matches g'(x) / g(x), its antiderivative is exactly ln|g(x)| + C.
- Trigonometric cotangent integrates to ln|sin x| because cos x is the derivative of sin x.
- Constant numerators over quadratics evaluate to inverse tangents, not natural logarithms.
- Plugging the original x boundaries into u(x) converts them into limits for the variable u.
- Transforming limits immediately removes any need to return to the original variable x.
- Evaluating the antiderivative directly at the new u-limits avoids double-evaluation errors.
- LIATE prioritizes Logarithmic, Inverse trig, Algebraic, Trig, then Exponential for picking u.
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives Techniques of Integration the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
If your differential gives du = 2x dx, what can you swap out for x dx?
Get it right to open this lesson and 23 more in the app.
How to learn Techniques of Integration, step by step
Go in order. 24 short levels across 3 sections, about 48 minutes in all, five minutes a day.
- 1
The Workhorse Methods in Action
- Smarter Substitution Strategies
- Integration by Parts and Cyclic Integrals
- 2
Algebraic and Geometric Transforms
- Trigonometric Powers and Product Identities
- Trigonometric Substitution for Radical Forms
- Partial Fractions for Rational Functions
- 3
Strategic Diagnosis and Harder Problems
- Method Diagnosis: Classifying Integrands
- Definite Integrals and Boundary Traps
- Multi-Step Solutions and Error Recovery
3 sections · 8 units · 24 levels. Built to play, not to enroll.
Questions about Techniques of Integration
How do I learn Techniques of Integration?
Go in order: start with The Workhorse Methods in Action, then Algebraic and Geometric Transforms, then Strategic Diagnosis and Harder Problems. On Tomo that's 24 short levels, about 48 minutes in all, played five minutes a day. Start free and play the first level in under a minute.
Is there a Duolingo-style app for Techniques of Integration?
Yes. Tomo works like Duolingo: short daily lessons, streaks and quick quizzes. Techniques of Integration is one of its free courses, and you can type any other topic to get a course of your own.
Is the Techniques of Integration course free?
Yes. Techniques of Integration is playable on Tomo's free plan, which doesn't expire. No credit card, no waitlist. Download the app and start your first lesson.
How long does it take to learn Techniques of Integration?
24 levels across 3 sections, about 48 minutes of play in total. It's built for five minutes a day, so you finish it a little at a time instead of all at once.
Do I need any background to start Techniques of Integration?
No. It's pitched at an intermediate level, so it starts where you are and builds up. If you can tap a phone, you can start.
Tomo is in ChatGPT
Tomo is also a ChatGPT plugin: the same playable courses, built and played inside your chat.
Add Tomo to ChatGPT- What to ask
- “Make me a Tomo course on black holes.” “Turn this chat into a Tomo course.”
- What you get
- Quick questions with instant feedback, hearts and stars, ready to play in about a minute.
- Where it goes
- No sign-up to start. Connect your Tomo account and your streak carries on in the app.
You pick the voice
Techniques of Integration is taught in the Explain Like I'm 5 style: no big words. promise.. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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