Integrals

If derivatives answer "How fast is something changing?", integrals answer "What is the total amount accumulated?" An integral is the reverse of a derivative—the inverse operation. While a derivative takes a function apart to find its rate of change, an integral puts pieces together to find the accumulated total. Think of it this way: if the derivative tells you the instantaneous velocity, the integral tells you the total distance traveled. This chapter introduces antiderivatives (the reverse of derivatives), develops the techniques of indefinite integration (finding families of antiderivatives), and establishes the Fundamental Theorem of Calculus, which connects integration to chapter-06-application-of-derivatives.

Understanding Antiderivatives and Indefinite Integrals

An antiderivative of f(x) is a function F(x) such that F'(x) = f(x). For example:

• Antiderivative of 2x is x² (since d/dx[x²] = 2x)
• Antiderivative of cos(x) is sin(x) (since d/dx[sin(x)] = cos(x))

But here's the key insight: if F(x) is an antiderivative of f(x), so is F(x) + C for any constant C:

• d/dx[x² + 5] = 2x
• d/dx[x² - 100] = 2x
• Both are antiderivatives of 2x

The indefinite integral of f(x), denoted ∫f(x)dx, represents the family of all antiderivatives:

∫f(x)dx = F(x) + C

where F'(x) = f(x) and C is the constant of integration.

Why the constant? Differentiating any constant gives zero, so we lose information about constant terms when taking derivatives. Integration recovers them as an arbitrary constant.

Building from Class 11 and Chapter 5

In Class 11, you computed derivatives using the power rule, product rule, chain rule, and more. Now we reverse those processes. From chapter-05-continuity-and-differentiability, you know that continuous functions have derivatives. Now we learn that differentiable functions have antiderivatives (though finding them analytically isn't always possible).

Basic Integration Formulas

These reverse the differentiation rules you learned:

Power Rule: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C (for n ≠ -1)

Special case: ∫(1/x) dx = ln|x| + C

Exponential:

• ∫eˣ dx = eˣ + C
• ∫aˣ dx = aˣ/ln(a) + C

Trigonometric:

• ∫sin(x) dx = -cos(x) + C
• ∫cos(x) dx = sin(x) + C
• ∫sec²(x) dx = tan(x) + C

Inverse Trig (from chapter-02-inverse-trigonometric-functions):

• ∫1/√(1 - x²) dx = arcsin(x) + C
• ∫1/(1 + x²) dx = arctan(x) + C

Integration Techniques

Substitution (u-substitution)

For integrals of the form ∫f(g(x))·g'(x) dx, substitute u = g(x), du = g'(x) dx:

∫f(u) du = F(u) + C = F(g(x)) + C

This reverses the chain rule.

Example: ∫x·e^(x²) dx

• Let u = x², du = 2x dx, so x dx = du/2
• ∫x·e^(x²) dx = ∫e^u · (du/2) = (1/2)e^u + C = (1/2)e^(x²) + C

Integration by Parts

For products, use ∫u dv = uv - ∫v du. Choose u and dv strategically using the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential functions take priority as u).

Example: ∫x·sin(x) dx

• u = x, dv = sin(x) dx
• du = dx, v = -cos(x)
• ∫x·sin(x) dx = -x·cos(x) - ∫(-cos(x)) dx = -x·cos(x) + sin(x) + C

Partial Fractions (for rational functions)

Express a rational function as a sum of simpler fractions to integrate term-by-term.

The Fundamental Theorem of Calculus

This theorem is the bridge between indefinite and definite integrals:

Theorem: If f is continuous on [a, b] and F is an antiderivative of f, then:

∫ₐᵇ f(x) dx = F(b) - F(a)

Meaning: The definite integral (area under the curve from a to b) equals the change in any antiderivative from a to b.

Notation: We write F(b) - F(a) as F(x)|ₐᵇ

Example: ∫₀¹ x² dx = (x³/3)|₀¹ = 1/3 - 0 = 1/3

Definite Integrals and Area

The definite integral ∫ₐᵇ f(x) dx represents the signed area between the curve f(x) and the x-axis from x = a to x = b:

• If f(x) > 0: area is positive
• If f(x) < 0: area is negative
• Net area accounts for both

To find total area regardless of sign: ∫ₐᵇ |f(x)| dx

Connections to Other Topics

• chapter-06-application-of-derivatives: Inverse operation; recovery from rates of change
• chapter-08-application-of-integrals: Using integrals to calculate areas and volumes
• chapter-09-differential-equations: Integration solves differential equations
• chapter-05-continuity-and-differentiability: Continuity ensures integrability

Socratic Questions

1. Why must we include a constant of integration C when finding the indefinite integral ∫f(x) dx, but not when computing the definite integral ∫ₐᵇ f(x) dx? What happens to the constant when you subtract F(a) from F(b)?

1. The chain rule for derivatives says d/dx[f(g(x))] = f'(g(x))·g'(x). How does u-substitution reverse this rule? Why do we need the factor g'(x) present in the integrand for substitution to work cleanly?

1. Explain the geometric meaning of the Fundamental Theorem of Calculus using a concrete example. How does computing F(b) - F(a) relate to finding the area under a curve?

1. For the integral ∫x·eˣ dx, would you choose u = x and dv = eˣ dx, or u = eˣ and dv = x dx? Why does the LIATE rule guide this choice, and how would the other choice lead to difficulties?

1. If you want to find the total area enclosed between two curves f(x) and g(x) from x = a to x = b, how does the integral ∫ₐᵇ |f(x) - g(x)| dx capture this? What would happen if you forgot the absolute value?