Integration — AI Study Guide

Mastering Integration

Integration is the reverse process of differentiation and represents accumulated change. The indefinite integral ∫f(x)dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration. The definite integral ∫_a^b f(x)dx represents the net area between f(x) and the x-axis on [a,b] — positive where f > 0, negative where f < 0. The Fundamental Theorem of Calculus (FTC) connects derivatives and integrals.

The Fundamental Theorem of Calculus has two parts. Part 1: if F(x) = ∫_a^x f(t)dt, then F'(x) = f(x) — the derivative of the integral of f recovers f. Part 2: ∫_a^b f(x)dx = F(b) - F(a), where F is any antiderivative of f — computing definite integrals requires finding antiderivatives. Together, they establish that differentiation and integration are inverse operations.

Integration techniques extend the basic antiderivatives. U-substitution (the chain rule in reverse): substitute u = g(x) to simplify the integral. Integration by parts (the product rule in reverse): ∫u dv = uv - ∫v du — use when the integrand is a product and one factor becomes simpler when differentiated. Trigonometric substitution handles integrands with a² - x², a² + x², or x² - a² by substituting x = a·sinθ, a·tanθ, or a·secθ, respectively.

Applications of definite integrals extend integration to practical problems. Area between curves: ∫_a^b [f(x) - g(x)]dx when f(x) ≥ g(x). Volume of revolution: disk/washer method or shell method. Arc length: ∫_a^b √(1 + [f'(x)]²)dx. Work done by a variable force. Average value of a function: (1/(b-a))∫_a^b f(x)dx. These applications appear throughout calculus courses and engineering.

Frequently Asked Questions: Integration

What is the Fundamental Theorem of Calculus?

The FTC has two parts: Part 1: d/dx[∫_a^x f(t)dt] = f(x). The derivative of the integral function recovers the integrand. Part 2: ∫_a^b f(x)dx = F(b) - F(a), where F'(x) = f(x). To evaluate a definite integral, find any antiderivative F and evaluate it at the endpoints. The FTC establishes differentiation and integration as inverse operations and is why we can evaluate definite integrals using antiderivatives rather than computing limits of Riemann sums.

When should I use u-substitution vs integration by parts?

U-substitution works when the integrand contains a composite function f(g(x)) and its derivative g'(x): let u = g(x), du = g'(x)dx. Use it when you see a function and (approximately) its derivative multiplied together. Integration by parts works when the integrand is a product of two functions — particularly when one becomes simpler upon differentiation and the other has a known antiderivative. LIATE mnemonic for which factor to call u: Logarithms, Inverse trig, Algebraic (polynomials), Trigonometric, Exponentials — earlier in the list = better choice for u.

How does Clario help with integration?

Clario processes your integration notes to generate flashcards covering integration rules, techniques, and the FTC, an AI summary of integration methods and applications, and practice problems from your specific calculus course material.

Study Calculus with Clario

Upload your Calculus course notes and Clario generates a complete study system — summaries, flashcards, practice quizzes, and exam prep — in under 60 seconds. Every study tool is built from your specific notes, not generic content.

Try Clario Free

No credit card required. 3 free study packs.