Integration made step-by-step with an AI tutor

Integration is pattern recognition under pressure. The hard part is choosing the technique, not doing the algebra. Noeta drills the choice itself — substitution vs parts vs partial fractions — until picking the right route is automatic.

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What this covers

Antiderivatives and the fundamental theorem
u-substitution
Integration by parts
Definite integrals and area under a curve
Partial fractions and trig integrals

Worked example

Problem

Evaluate ∫ 2x·e^(x²) dx

1
Look for an inner function
The exponent x² has derivative 2x, which already appears as a factor — a substitution signal.
2
Substitute
Let u = x², so du = 2x dx. The integral becomes ∫ e^u du.
3
Integrate
∫ e^u du = e^u + C.
4
Back-substitute
Replace u with x².

Answer: e^(x²) + C

Mistakes that cost marks

Forgetting the constant of integration on indefinite integrals.
Substituting u but leaving dx in the integral.
Not changing the limits when substituting in a definite integral.

Questions students ask

How do I know when to use substitution or parts?

Substitution when one factor is the derivative of another part; parts when you have a product of unrelated families such as x·ln x. Noeta quizzes you on that decision specifically.

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