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Fundamental Theorem
of Calculus

Master the powerful connection between differentiation and integration: how definite integrals are evaluated and why accumulation functions have derivatives.

AP Calculus BC

What Is a Definite Integral?

A definite integral computes the net signed area between a function \( f(x) \) and the \( x \)-axis over an interval \( [a, b] \). It is written as:

Area above the \( x \)-axis counts as positive, and area below counts as negative. The result is a single number, not a function.

Visualizing Accumulation

x y a b Area
\(f(x)\)

Why Does This Matter?

Before the Fundamental Theorem, evaluating definite integrals required computing tedious Riemann sums: partitioning the interval into infinitely many rectangles. The FTC provides a shortcut: just find an antiderivative.

Without FTC
Evaluate \( \lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\Delta x \): slow, complex, often impossible in closed form.
With FTC
Find \( F(x) \) where \( F'(x)=f(x) \), then compute \( F(b)-F(a) \): fast, elegant, exact.

FTC Part 1: The Accumulation Function

In words: The derivative of the integral (with a variable upper limit) gives back the original function. Differentiation and integration are inverse operations.

FTC Part 2: The Evaluation Theorem

This is the evaluation shortcut. Instead of limits of sums, find any function \( F \) with \( F'=f \), plug in the endpoints, and subtract.

\(F(x)\)
 a b
\(F(a)\)
\(F(b)\)
 gap

Part 1 vs Part 2: Key Distinction

FTC Part 1 FTC Part 2
Statement \( \frac{d}{dx}\!\int_a^x f(t)\,dt = f(x) \) \( \int_a^b f(x)\,dx = F(b)-F(a) \)
Direction Integration → Differentiation Differentiation → Integration
Output A function A number
Key idea Derivative undoes integral Antiderivative evaluates integral
Common use Finding rates of accumulation Computing definite integrals

Chain Rule Extension of FTC Part 1

When the upper limit is not simply \( x \) but a function \( g(x) \), apply the chain rule:

1
Identify the upper limit \( g(x) \) and compute its derivative \( g'(x) \).
2
Replace \( t \) with \( g(x) \) in the integrand \( f(t) \) to get \( f(g(x)) \).
3
Multiply by \( g'(x) \) to complete the chain rule.

How to Apply FTC Part 2

1
Check continuity: Confirm \( f(x) \) is continuous on \( [a,b] \). If not, the FTC does not directly apply.
2
Find an antiderivative: Determine \( F(x) \) such that \( F'(x) = f(x) \). No need for \( +C \) here.
3
Evaluate at the bounds: Compute \( F(b) \) and \( F(a) \).
4
Subtract: The answer is \( F(b) - F(a) \). Remember: upper minus lower.

Example 1: Evaluating \(\int_1^4 x^2\,dx\)

1 2 3 4
\(x^2\)
1
Antiderivative: \( F(x) = \frac{x^3}{3} \)
2
Evaluate: \( F(4) - F(1) = \frac{64}{3} - \frac{1}{3} = \frac{63}{3} = \) 21

Example 2: Evaluating \(\int_0^{\pi} \sin x\,dx\)

0
\(\frac{\pi}{2}\)
\(\pi\)
\(\sin x\)
 1
1
Antiderivative of \( \sin x \): \( F(x) = -\cos x \)
2
Evaluate: \( -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = \) 2

Example 3: FTC Part 1 with Chain Rule

Problem: Find \( \frac{d}{dx}\!\int_2^{x^3} e^{-t^2}\,dt \).

1
Identify the pieces: Upper limit \( g(x)=x^3 \), integrand \( f(t)=e^{-t^2} \).
2
Apply FTC Part 1 + chain rule: Replace \( t \) with \( g(x) \): \( f(g(x)) = e^{-(x^3)^2} = e^{-x^6} \).
3
Multiply by \( g'(x) = 3x^2 \):

Notice: we never needed to find a closed-form antiderivative of \( e^{-t^2} \) (which doesn't exist in elementary functions). FTC Part 1 bypasses integration entirely.

Both Limits Variable?

When both the lower and upper limits depend on \( x \), split the integral at a constant \( c \):

Then differentiate each piece using FTC Part 1 with the chain rule:

Key Properties of Definite Integrals

Reversing Limits
\( \int_a^b f\,dx = -\int_b^a f\,dx \)
Zero Width
\( \int_a^a f\,dx = 0 \)
Additivity
\( \int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx \)
Constant Multiple
\( \int_a^b k\cdot f\,dx = k\!\int_a^b f\,dx \)
Sum Rule
\( \int_a^b (f+g)\,dx = \int_a^b f\,dx + \int_a^b g\,dx \)

Common Mistakes to Avoid

✕ Forgetting the chain rule
When the upper limit is \( g(x) \neq x \), you must multiply by \( g'(x) \). Writing \( f(g(x)) \) alone is incomplete.
✕ Wrong subtraction order
It is \( F(b)-F(a) \), not \( F(a)-F(b) \). Reversing gives the negative of the correct answer.
✕ Adding \( +C \)
Definite integrals yield a number. The constant cancels: \( (F(b)+C)-(F(a)+C) = F(b)-F(a) \).
✕ Ignoring continuity
FTC requires \( f \) to be continuous on \( [a,b] \). If \( f \) has a discontinuity in the interval, the theorem does not apply directly.

Summary: FTC at a Glance

Concept Formula Key Detail
FTC Part 1 \( \frac{d}{dx}\!\int_a^x f(t)\,dt = f(x) \) Output is a function
FTC Part 1 + Chain \( f(g(x))\cdot g'(x) \) Upper limit \( = g(x) \)
FTC Part 2 \( \int_a^b f\,dx = F(b)-F(a) \) Output is a number
Both limits variable Split at constant, apply Part 1 twice Subtract lower limit term
Prerequisite \( f \) continuous on \( [a,b] \) Always verify first

Question 1 of 5

Evaluate \( \displaystyle\int_0^3 (2x+1)\,dx \).

Question 2 of 5

Find \( \displaystyle\frac{d}{dx}\!\int_1^x \ln(t)\,dt \).

Question 3 of 5

Find \( \displaystyle\frac{d}{dx}\!\int_0^{x^2} \cos(t)\,dt \).

Question 4 of 5

Evaluate \( \displaystyle\int_1^e \frac{1}{x}\,dx \).

Question 5 of 5

If \( \displaystyle\int_0^5 f(x)\,dx = 7 \) and \( \displaystyle\int_0^3 f(x)\,dx = 4 \), what is \( \displaystyle\int_3^5 f(x)\,dx \)?