Integration Basics

Grade 12 · Math · Study Guide

An introduction to antiderivatives, definite integrals, the Fundamental Theorem of Calculus, and core integration techniques.

Integration is the reverse process of differentiation. An antiderivative (or indefinite integral) of f(x) is any function F(x) such that F'(x) = f(x). We write ∫f(x)dx = F(x) + C, where C is the constant of integration.

The Power Rule for integration states ∫xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ -1. Other key antiderivatives include ∫eˣ dx = eˣ + C, ∫(1/x) dx = ln|x| + C, ∫cos x dx = sin x + C, and ∫sin x dx = -cos x + C.

The Fundamental Theorem of Calculus (FTC) connects differentiation and integration. Part 1 states that if F(x) = ∫ₐˣ f(t) dt, then F'(x) = f(x). Part 2 states that ∫ₐᵇ f(x) dx = F(b) - F(a), allowing exact evaluation of definite integrals. The definite integral ∫ₐᵇ f(x) dx represents the net signed area between the curve and the x-axis from x=a to x=b.

U-substitution is the main technique for integrating composite functions, analogous to the Chain Rule in differentiation. Let u = g(x), then du = g'(x) dx, transforming the integral into a simpler form.

Key terms

Antiderivative
A function F(x) whose derivative is f(x); also called an indefinite integral
Definite Integral
∫ₐᵇ f(x) dx; represents net signed area under f from a to b, evaluated as F(b) - F(a)
Constant of Integration
The +C added to every indefinite integral, representing the family of all antiderivatives
Fundamental Theorem of Calculus
Connects differentiation and integration; Part 2 states ∫ₐᵇ f(x)dx = F(b) - F(a)
U-Substitution
An integration technique where a substitution u = g(x) simplifies a composite integrand
Net Signed Area
The value of a definite integral; area above x-axis counts positive, area below counts negative

Flashcards

∫xⁿ dx
xⁿ⁺¹/(n+1) + C, provided n ≠ -1
∫eˣ dx
eˣ + C
∫cos x dx
sin x + C
∫sin x dx
-cos x + C
∫(1/x) dx
ln|x| + C
U-Substitution Steps
1) Choose u = g(x). 2) Compute du = g'(x)dx. 3) Rewrite integral in terms of u. 4) Integrate. 5) Substitute back.
FTC Part 2
∫ₐᵇ f(x)dx = F(b) - F(a), where F is any antiderivative of f
Area Between Curves
∫ₐᵇ [f(x) - g(x)] dx where f(x) ≥ g(x) on [a,b]

Questions

  1. Evaluate ∫6x² dx.

    1. 2x³ + C
    2. 12x + C
    3. 3x² + C
    4. 6x³ + C
    Show answer

    Answer: 2x³ + C

    Power Rule for integration: ∫6x² dx = 6 · x³/3 + C = 2x³ + C.

  2. The Fundamental Theorem of Calculus Part 2 states that ∫ₐᵇ f(x)dx = F(b) ___ F(a).

    Show answer

    Answer: minus

    FTC Part 2: ∫ₐᵇ f(x)dx = F(b) - F(a). You evaluate the antiderivative at the upper bound and subtract its value at the lower bound.

  3. Using u-substitution, evaluate ∫2x(x² + 1)³ dx.

    1. (x² + 1)⁴ / 2 + C
    2. (x² + 1)⁴ / 4 + C
    3. x²(x² + 1)⁴ + C
    4. 8x²(x² + 1)² + C
    Show answer

    Answer: (x² + 1)⁴ / 4 + C

    Let u = x²+1, du = 2x dx. Integral becomes ∫u³ du = u⁴/4 + C = (x²+1)⁴/4 + C.

  4. The definite integral ∫₀¹ x² dx equals 1/3.

    Show answer

    Answer: True

    ∫₀¹ x² dx = [x³/3]₀¹ = 1/3 - 0 = 1/3.

  5. What is ∫(1/x) dx?

    1. x⁻² + C
    2. -x⁻² + C
    3. ln|x| + C
    4. 1 + C
    Show answer

    Answer: ln|x| + C

    ∫(1/x) dx = ln|x| + C. The Power Rule fails for n = -1; this integral gives a logarithm.

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