Calculus I · Unit 3A · review

Unit 3A Review: Integral Foundations and Techniques

Unit 3A Review

Summary

Core ideas

• Antiderivatives reverse differentiation and form families differing by constants. • Riemann sums approximate continuously distributed totals. • Definite integrals are signed accumulations and limits of Riemann sums. • The Fundamental Theorem links differentiation and integration. • Substitution reverses the chain rule; integration by parts reverses the product rule. • Numerical methods estimate integrals when exact antiderivatives are unavailable. • Improper integrals are defined by limits and may converge or diverge.

Exercise

Find $\int(3x^2-4/x+\cos x)dx$ .

Exercise

Approximate $\int_0^2e^{-x^2}dx$ using four midpoint rectangles.

Exercise

Differentiate $F(x)=\int_1^{x^3}\ln(1+t^2)dt$ .

Exercise

Evaluate $\int_0^1x(1+x^2)^5dx$ .

Exercise

Evaluate $\int x\cos x\,dx$ .

Exercise

Evaluate $\int\frac{3x+5}{x^2+5x+6}dx$ by partial fractions.

Exercise

Determine convergence of $\int_2^\infty x^{-3/2}dx$ .

After the explanation

Use the section idea

Reading lens

Mixed integral work tests recognition: classify the output and structure before committing to a method.

Mental model

A complete response carries setup, method, computation, bounds or constants, units, interpretation, and an independent verification.

Decision

Attempt the entire problem first, then use one answer at a time to locate the earliest reasoning decision that needs repair.

Common trap

Reading a key before modeling the problem or treating every mismatch as algebra when the first error was conceptual.

Check yourself

Can you reproduce the reasoning without the key and explain why the final form fits the question?

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.