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Home/Calculus/Chapter 6/Fundamental Theorem of Calculus

CALC-6.3

4-5 hours

Fundamental Theorem of Calculus

The most important theorem in calculus: the Newton-Leibniz formula reveals that differentiation and integration are inverse operations.

Learning Objectives

Define and analyze variable upper limit integrals

Prove continuity of the integral function F(x)

Establish differentiability: F'(x) = f(x)

Prove the Newton-Leibniz formula

Apply FTC to compute definite integrals

Differentiate integrals with variable limits

Evaluate limits using L'Hôpital with integrals

1. Variable Upper Limit Integral

Definition 6.8: Variable Upper Limit Integral

If $f \in R[a, b]$ , define the integral function:

F(x) = \int_a^x f(t)\,dt, \quad x \in [a, b]

Note: We use a different variable $t$ inside to avoid confusion with the upper limit $x$ .

Theorem 6.16: Continuity of the Integral Function

If $f \in R[a, b]$ , then $F(x) = \int_a^x f(t)\,dt$ is continuous on $[a, b]$ .

In fact, $F$ is Lipschitz continuous: $|F(x) - F(y)| \leq M|x - y|$ where $M = \sup|f|$ .

Proof of Theorem 6.16:

For $x, y \in [a, b]$ with $x < y$ :

|F(y) - F(x)| = \left| \int_x^y f(t)\,dt \right| \leq \int_x^y |f(t)|\,dt \leq M(y - x)

This shows $F$ is Lipschitz continuous, hence uniformly continuous.

∎

2. FTC Part 1: Derivative of the Integral

Theorem 6.17: Fundamental Theorem of Calculus (Part 1)

Let $f \in R[a, b]$ and $F(x) = \int_a^x f(t)\,dt$ .

If $f$ is continuous at $x_0 \in [a, b]$ , then $F$ is differentiable at $x_0$ and:

F'(x_0) = f(x_0)

In short: “The derivative of an integral is the integrand.”

Proof of Theorem 6.17:

Consider the difference quotient for $h > 0$ :

\frac{F(x_0 + h) - F(x_0)}{h} = \frac{1}{h} \int_{x_0}^{x_0+h} f(t)\,dt

Since $f$ is continuous at $x_0$ , for any $\epsilon > 0$ , there exists $\delta > 0$ such that:

$|t - x_0| < \delta \Rightarrow |f(t) - f(x_0)| < \epsilon$

For $0 < h < \delta$ :

\left| \frac{F(x_0 + h) - F(x_0)}{h} - f(x_0) \right| = \left| \frac{1}{h} \int_{x_0}^{x_0+h} [f(t) - f(x_0)]\,dt \right|

\leq \frac{1}{h} \int_{x_0}^{x_0+h} |f(t) - f(x_0)|\,dt < \frac{1}{h} \cdot \epsilon \cdot h = \epsilon

Similarly for $h < 0$ . Thus $F'(x_0) = f(x_0)$ .

∎

Corollary 6.4: Continuous Functions Have Antiderivatives

If $f \in C[a, b]$ , then $F(x) = \int_a^x f(t)\,dt$ is an antiderivative of $f$ on $[a, b]$ .

Remark 6.5: Antiderivatives vs. Integrability

Important distinctions:

An integrable function may NOT have an elementary antiderivative (e.g., $e^{-x^2}$ )
A function with an antiderivative may NOT be integrable (pathological examples exist)
For continuous functions, both directions work!

3. Newton-Leibniz Formula (FTC Part 2)

Theorem 6.18: Newton-Leibniz Formula

Let $f \in R[a, b]$ and suppose $F$ is an antiderivative of $f$ on $[a, b]$ (i.e., $F'(x) = f(x)$ ). Then:

\int_a^b f(x)\,dx = F(b) - F(a) =: F(x)\Big|_a^b

Proof of Theorem 6.18:

Take any partition $\Delta: a = x_0 < x_1 < \cdots < x_n = b$ .

By the Mean Value Theorem, for each $k$ there exists $\xi_k \in (x_{k-1}, x_k)$ such that:

F(x_k) - F(x_{k-1}) = F'(\xi_k)(x_k - x_{k-1}) = f(\xi_k)\Delta x_k

Summing over all $k$ (telescoping sum):

F(b) - F(a) = \sum_{k=1}^n [F(x_k) - F(x_{k-1})] = \sum_{k=1}^n f(\xi_k)\Delta x_k

Taking $\|\Delta\| \to 0$ , the right side converges to $\int_a^b f(x)\,dx$ .

∎

Remark 6.6: Evaluation Notation

The notation $F(x)\big|_a^b$ or $[F(x)]_a^b$ means:

F(x)\Big|_a^b = F(b) - F(a)

This is read as “ $F(x)$ evaluated from $a$ to $b$ ”.

Example 6.12: Basic Application of Newton-Leibniz

Compute: $\int_0^1 x^2\,dx$

Antiderivative: $F(x) = \frac{x^3}{3}$

\int_0^1 x^2\,dx = \frac{x^3}{3}\Big|_0^1 = \frac{1}{3} - 0 = \frac{1}{3}

4. Differentiation with Variable Limits

Corollary 6.5: General Leibniz Rule

If $\phi(x)$ and $\psi(x)$ are differentiable and $f$ is continuous:

\frac{d}{dx} \int_{\phi(x)}^{\psi(x)} f(t)\,dt = f(\psi(x))\psi'(x) - f(\phi(x))\phi'(x)

Proof of Corollary 6.5:

Let $G(u) = \int_a^u f(t)\,dt$ . Then:

\int_{\phi(x)}^{\psi(x)} f(t)\,dt = G(\psi(x)) - G(\phi(x))

By the chain rule:

\frac{d}{dx}[G(\psi(x)) - G(\phi(x))] = G'(\psi(x))\psi'(x) - G'(\phi(x))\phi'(x)

Since $G'(u) = f(u)$ , the result follows.

∎

Example 6.13: Variable Limits

Find: $\frac{d}{dx} \int_x^{x^2} e^{-t^2}\,dt$

Here $\phi(x) = x$ , $\psi(x) = x^2$ , $f(t) = e^{-t^2}$ .

\frac{d}{dx} \int_x^{x^2} e^{-t^2}\,dt = e^{-x^4} \cdot 2x - e^{-x^2} \cdot 1 = 2xe^{-x^4} - e^{-x^2}

Example 6.14: Limit Evaluation

Evaluate: $\lim_{x \to 0} \frac{\int_0^x \cos t^2\,dt}{x}$

This is $\frac{0}{0}$ form. By L'Hôpital's Rule:

\lim_{x \to 0} \frac{\int_0^x \cos t^2\,dt}{x} = \lim_{x \to 0} \frac{\cos x^2}{1} = \cos 0 = 1

5. Applications and Examples

Example 6.15: Converting Sums to Integrals

Evaluate: $\lim_{n \to \infty} n\left( \frac{1}{n^2+1} + \frac{1}{n^2+4} + \cdots + \frac{1}{n^2+n^2} \right)$

= \lim_{n \to \infty} \sum_{k=1}^n \frac{n}{n^2 + k^2} = \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{1 + (k/n)^2} \cdot \frac{1}{n}

This is a Riemann sum for $f(x) = \frac{1}{1+x^2}$ on $[0, 1]$ :

= \int_0^1 \frac{dx}{1+x^2} = \arctan x\Big|_0^1 = \frac{\pi}{4}

Example 6.16: Uniform Convergence Application

Problem: If $f \in C^1[0, 1]$ , prove:

\lim_{n \to \infty} \sum_{k=1}^n \left[ f\left( x + \frac{k}{n^2+k^2} \right) - f(x) \right] = f'(x) \cdot \frac{\ln 2}{2}

Sketch:

Use Mean Value Theorem and recognize the Riemann sum structure for $\int_0^1 \frac{dt}{1+t^2} = \frac{\pi}{4}$ ... (detailed proof involves careful analysis).

6. Common Mistakes

❌ Wrong: $\frac{d}{dx}\int_a^b f(t)\,dt = f(x)$

The integral from $a$ to $b$ (both constants) is a constant. Its derivative is 0!

❌ Forgetting the Chain Rule

$\frac{d}{dx}\int_a^{x^2} f = f(x^2) \cdot 2x$ , not just $f(x^2)$ !

❌ Wrong sign for lower limit

$\frac{d}{dx}\int_x^b f = -f(x)$ (note the minus sign!)

❌ Assuming all integrals have elementary antiderivatives

$\int e^{-x^2}dx$ has no elementary form. The integral exists but can't be expressed simply.

7. Additional Examples

Example 6.17: Computing Definite Integrals

Evaluate: $\int_0^{\pi/2} \cos x\,dx$

Solution:

Antiderivative of $\cos x$ is $\sin x$ .

\int_0^{\pi/2} \cos x\,dx = \sin x\Big|_0^{\pi/2} = \sin\frac{\pi}{2} - \sin 0 = 1 - 0 = 1

Example 6.18: Polynomial Integration

Evaluate: $\int_1^2 (3x^2 - 2x + 1)\,dx$

Solution:

Antiderivative: $F(x) = x^3 - x^2 + x$

\int_1^2 (3x^2 - 2x + 1)\,dx = [x^3 - x^2 + x]_1^2

= (8 - 4 + 2) - (1 - 1 + 1) = 6 - 1 = 5

Example 6.19: Exponential and Logarithmic

Evaluate: $\int_1^e \frac{1}{x}\,dx$ and $\int_0^1 e^x\,dx$

Solutions:

\int_1^e \frac{1}{x}\,dx = \ln x\Big|_1^e = \ln e - \ln 1 = 1 - 0 = 1

\int_0^1 e^x\,dx = e^x\Big|_0^1 = e - 1

Example 6.20: Differentiation Under Integral Sign

Find: $F'(x)$ where $F(x) = \int_0^{x^3} \sin(t^2)\,dt$

Solution:

Using the Leibniz rule with $\psi(x) = x^3$ :

F'(x) = \sin((x^3)^2) \cdot \frac{d}{dx}(x^3) = \sin(x^6) \cdot 3x^2 = 3x^2 \sin(x^6)

Example 6.21: Both Limits Variable

Find: $\frac{d}{dx} \int_{\sin x}^{\cos x} e^{t^2}\,dt$

Solution:

Using the Leibniz rule:

= e^{\cos^2 x} \cdot (-\sin x) - e^{\sin^2 x} \cdot \cos x

= -\sin x \cdot e^{\cos^2 x} - \cos x \cdot e^{\sin^2 x}

Example 6.22: L'Hôpital with Integrals

Evaluate: $\lim_{x \to 0} \frac{\int_0^x \sin t^2\,dt}{x^3}$

Solution:

This is $\frac{0}{0}$ form. Apply L'Hôpital:

\lim_{x \to 0} \frac{\sin x^2}{3x^2}

Still $\frac{0}{0}$ . Apply again or use Taylor: $\sin x^2 \approx x^2$ for small $x$ :

= \lim_{x \to 0} \frac{x^2}{3x^2} = \frac{1}{3}

Example 6.23: Finding Functions from Integral Equations

Problem: Find $f(x)$ if $\int_0^x f(t)\,dt = x^2 + x\sin x$

Solution:

By FTC Part 1, differentiate both sides:

f(x) = \frac{d}{dx}(x^2 + x\sin x) = 2x + \sin x + x\cos x

Verify: $\int_0^0 f(t)\,dt = 0$ and RHS at $x=0$ is $0 + 0 = 0$ . ✓

8. Practice Problems

Problem Set A: Newton-Leibniz Formula

1.Evaluate $\int_0^{\pi} \sin x\,dx$
2.Evaluate $\int_1^4 \sqrt{x}\,dx$
3.Evaluate $\int_0^1 \frac{1}{1+x^2}\,dx$
4.Evaluate $\int_{-1}^1 |x|\,dx$

Problem Set B: Differentiation of Integrals

5.Find $\frac{d}{dx}\int_1^x \ln t\,dt$
6.Find $\frac{d}{dx}\int_0^{x^2} e^{-t^2}\,dt$
7.Find $\frac{d}{dx}\int_x^{2x} \frac{1}{t}\,dt$
8.Find $\frac{d}{dx}\int_{\sqrt{x}}^{x^2} \cos(t^3)\,dt$

Problem Set C: Limits and Applications

9.Evaluate $\lim_{x \to 0} \frac{\int_0^x e^{t^2}\,dt}{x}$
10.Evaluate $\lim_{x \to 0} \frac{\int_0^x \tan t\,dt}{x^2}$
11.If $F(x) = \int_0^x (x-t)f(t)\,dt$ , find $F''(x)$
12.Solve: $\int_0^x f(t)\,dt = x + \int_x^1 tf(t)\,dt$ for $f$

Remark 6.7: Answers to Selected Problems

Problem 1: $2$

Problem 2: $\frac{14}{3}$

Problem 3: $\frac{\pi}{4}$

Problem 5: $\ln x$

Problem 6: $2xe^{-x^4}$

Problem 9: $1$

9. Deeper Insights

Why FTC is Revolutionary

Unification

Before FTC, differentiation (tangent problem) and integration (area problem) were seen as separate. FTC reveals they are inverse operations—a profound insight.

Computational Power

Without FTC, computing $\int_0^1 x^{100}\,dx$ would require summing 100 terms and taking limits. With FTC: $\frac{x^{101}}{101}\big|_0^1 = \frac{1}{101}$ !

Existence of Antiderivatives

FTC Part 1 guarantees that every continuous function has an antiderivative—even if we can't write it in elementary form. The integral $F(x) = \int_a^x f$ always works!

Physical Interpretation

If $v(t)$ is velocity, then $\int_a^b v(t)\,dt = s(b) - s(a)$ is displacement. The rate of change ( $v = s'$ ) and cumulative change are intimately linked.

Theorem 6.19: Generalized FTC

If $F$ is absolutely continuous on $[a, b]$ , then $F'$ exists almost everywhere, $F' \in L^1[a,b]$ , and:

F(x) - F(a) = \int_a^x F'(t)\,dt

This is the Lebesgue version of FTC, valid for a broader class of functions.

Remark 6.8: Non-elementary Antiderivatives

Some important functions have no elementary antiderivative:

Error function

\text{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt

Sine integral

\text{Si}(x) = \int_0^x \frac{\sin t}{t}\,dt

Logarithmic integral

\text{li}(x) = \int_0^x \frac{dt}{\ln t}

Fresnel integral

S(x) = \int_0^x \sin(t^2)\,dt

10. Historical Context

Isaac Newton (1642-1727)

Developed his “method of fluxions” in the 1660s, recognizing that differentiation and integration are inverse processes. He kept his work largely unpublished until later.

Gottfried Wilhelm Leibniz (1646-1716)

Independently developed calculus in the 1670s-80s. His notation ( $\int$ , $dx$ ) and systematic approach proved more influential. The “Newton-Leibniz formula” honors both.

James Gregory (1638-1675)

Scottish mathematician who stated a geometric version of FTC before Newton and Leibniz, though his work was less complete and systematic.

Isaac Barrow (1630-1677)

Newton's teacher at Cambridge who proved a geometric form of FTC in his “Lectiones Geometricae” (1670), directly influencing Newton's work.

11. Summary of Key Results

Result	Formula	Conditions
FTC Part 1	$\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$	$f$ continuous at $x$
Newton-Leibniz	$\int_a^b f\,dx = F(b) - F(a)$	$F' = f$ , $f \in R[a,b]$
Leibniz Rule	$\frac{d}{dx}\int_\phi^\psi f = f(\psi)\psi' - f(\phi)\phi'$	$f, \phi, \psi$ differentiable
Continuity of $F$	$\|F(x) - F(y)\| \leq M\|x-y\|$	$f \in R[a,b]$ , $M = \sup\|f\|$

12. What's Next?

In the next section, we learn Computation Techniques for definite integrals:

Substitution method for definite integrals

Integration by parts formula

Wallis integrals and reduction formulas

Symmetry tricks for efficient computation

13. Study Tips

Memorize the Two Parts

FTC Part 1: Differentiation undoes integration. FTC Part 2 (Newton-Leibniz): How to evaluate definite integrals using antiderivatives.

Master the Leibniz Rule

For variable limits: differentiate the upper limit (multiply by its derivative), subtract the lower limit term. $\frac{d}{dx}\int_{\phi(x)}^{\psi(x)} f(t)\,dt = f(\psi)\psi' - f(\phi)\phi'$

Check Continuity

FTC Part 1 requires $f$ to be continuous at the point of differentiation. FTC Part 2 requires an antiderivative to exist.

Practice L'Hôpital Applications

Many limit problems involve $\frac{0}{0}$ forms with integrals. Use FTC to differentiate the numerator.

14. Challenge Problems

Challenge 1: Nested Integrals

Find $F''(x)$ where:

F(x) = \int_0^x \left(\int_0^t f(s)\,ds\right) dt

Answer: $F''(x) = f(x)$

Challenge 2: Integral Equation

Solve for $f$ :

f(x) = 1 + \int_0^x f(t)\,dt

Hint: Differentiate both sides and solve the ODE.

Challenge 3: Limit with Parameter

Evaluate:

\lim_{x \to 0^+} \frac{\int_0^x \sin(t^2)\,dt}{x^3}

Hint: Apply L'Hôpital twice and use Taylor expansion.

Challenge 4: Generalized Leibniz

For parameter-dependent integrands:

\frac{d}{dx}\int_a^b f(x, t)\,dt = \int_a^b \frac{\partial f}{\partial x}(x, t)\,dt

When does this hold? What conditions are needed?

15. Connections to Other Topics

Differential Equations

FTC is key to solving ODEs. $y' = f(x)$ has solution $y = \int f(x)\,dx + C$ . Many techniques rely on the inverse relationship between derivatives and integrals.

Physics Applications

Position from velocity: $s(t) = s_0 + \int_0^t v(\tau)\,d\tau$ . Work from force: $W = \int_a^b F(x)\,dx$ . FTC unifies these computations.

Probability Theory

CDF from PDF: $F(x) = \int_{-\infty}^x f(t)\,dt$ . By FTC: $F'(x) = f(x)$ —the PDF is the derivative of the CDF.

Complex Analysis

FTC generalizes to complex contour integrals. If $F' = f$ on a region, then $\int_\gamma f\,dz = F(\text{end}) - F(\text{start})$ .

16. Quick Reference Card

Key Formulas

FTC 1

\frac{d}{dx}\int_a^x f = f(x)

FTC 2

\int_a^b f = F(b) - F(a)

Leibniz

f(\psi)\psi' - f(\phi)\phi'

Common Patterns

Upper limit $g(x)$ :

\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)

Lower limit $h(x)$ :

\frac{d}{dx}\int_{h(x)}^b f(t)\,dt = -f(h(x)) \cdot h'(x)

Properties of $F(x)$

Continuous if $f \in R[a,b]$
Differentiable where $f$ is continuous
$F(a) = 0$

Conditions

FTC 1: $f$ continuous at $x$
FTC 2: $F' = f$ on $[a,b]$
Both: $f \in R[a,b]$

Common Errors

Forgetting chain rule
Wrong sign for lower limit
Assuming all $f$ have elementary $F$

17. Extended Examples

Example 6.24: Convolution-type Integral

Problem: Find $F'(x)$ where $F(x) = \int_0^x (x - t)^2 f(t)\,dt$

Solution:

Expand: $(x-t)^2 = x^2 - 2xt + t^2$

F(x) = x^2\int_0^x f(t)\,dt - 2x\int_0^x tf(t)\,dt + \int_0^x t^2 f(t)\,dt

Now differentiate using product rule + FTC:

F'(x) = 2x\int_0^x f + x^2 f(x) - 2\int_0^x tf\,dt - 2x \cdot xf(x) + x^2f(x)

= 2x\int_0^x f(t)\,dt - 2\int_0^x tf(t)\,dt

Example 6.25: Asymptotic Behavior

Problem: Find $\lim_{x \to \infty} \frac{\int_0^x e^{-t^2}\,dt}{\frac{e^{-x^2}}{2x}}$

Solution:

This is $\frac{\infty}{0}$ . Rewrite and apply L'Hôpital:

\lim_{x \to \infty} \frac{2x\int_0^x e^{-t^2}\,dt}{e^{-x^2}} = \lim \frac{2\int + 2x \cdot e^{-x^2}}{-2xe^{-x^2}}

The integral grows like $\sqrt{\pi}/2$ , so this limit is $-1$ .

Example 6.26: Area Under Parametric Curve

Problem: Find the area under $x = t^2$ , $y = t^3$ for $t \in [0, 1]$

Solution:

Use $A = \int y\,dx = \int_0^1 t^3 \cdot 2t\,dt$ :

A = \int_0^1 2t^4\,dt = \frac{2t^5}{5}\Big|_0^1 = \frac{2}{5}

Fundamental Theorem Quiz

Questions

Correct

Accuracy

The function

F(x) = \int_a^x f(t)\,dt

is called:

Easy

Not attempted

f \in R[a,b]

, then

F(x) = \int_a^x f(t)\,dt

is:

Easy

Not attempted

f

is continuous at

x_0

, then

F'(x_0)

equals:

Easy

Not attempted

The Newton-Leibniz formula states:

Easy

Not attempted

\frac{d}{dx}\int_a^{x^2} \sin t\,dt

equals:

Medium

Not attempted

\frac{d}{dx}\int_{x}^{x^2} f(t)\,dt

equals:

Hard

Not attempted

\lim_{x \to 0} \frac{\int_0^x \cos t^2\,dt}{x}

equals:

Medium

Not attempted

f \in R[a,b]

but NOT continuous, then

F(x) = \int_a^x f

is:

Hard

Not attempted

Frequently Asked Questions

Why is FTC so important?

FTC connects the two main operations of calculus: differentiation and integration are inverse operations. This lets us evaluate definite integrals using antiderivatives instead of limits of Riemann sums.

Does every integrable function have an antiderivative?

No! A function can be integrable without having an antiderivative expressible in elementary functions. Example: e^(-x²) is integrable but has no elementary antiderivative.

What if f is discontinuous?

If f ∈ R[a,b], F(x) = ∫ₐˣ f is still continuous. However, F is only differentiable at points where f is continuous, and F'(x₀) = f(x₀) only at such points.

Can the lower limit also vary?

Yes! Use ∫ₐˣ = -∫ˣₐ. For ∫_{g(x)}^{h(x)} f(t)dt, differentiate as: f(h(x))h'(x) - f(g(x))g'(x).

Why does the notation F(x)|ₐᵇ mean F(b) - F(a)?

This notation (evaluation bar) is shorthand. We 'evaluate at b' then 'subtract evaluation at a'. It comes from the Newton-Leibniz formula.

Key Takeaways

FTC Part 1

\frac{d}{dx}\int_a^x f(t)\,dt = f(x)

Newton-Leibniz

\int_a^b f\,dx = F(b) - F(a)

Leibniz Rule

\frac{d}{dx}\int_{\phi}^{\psi} f = f(\psi)\psi' - f(\phi)\phi'

Key Insight

Differentiation and integration are inverse operations!

Continuity of $F$

$F(x) = \int_a^x f$ is always continuous

Existence

Every continuous $f$ has an antiderivative via FTC

Decision Guide: Which Formula?

Evaluating: Newton-Leibniz with antiderivative

Differentiating: FTC Part 1 + chain rule

Variable limits: Leibniz rule

Limit problems: L'Hôpital + FTC

Integral equations: Differentiate both sides

No elementary form: Define

F

via integral

Quick Decision Guide

When to Use FTC Part 1

Differentiating integral with variable upper limit
Solving for functions from integral equations
Proving existence of antiderivatives

When to Use Newton-Leibniz

Computing definite integrals with known antiderivative
Evaluating areas, volumes, work
Finding net change from rate of change

18. Common Exam Patterns

Pattern 1: Direct Differentiation

Given $F(x) = \int_a^{g(x)} f(t)\,dt$ , find $F'(x)$ .

Solution: Apply chain rule: $F'(x) = f(g(x)) \cdot g'(x)$

Pattern 2: Limit Problems

Evaluate $\lim_{x \to a} \frac{\int_a^x f(t)\,dt}{g(x)}$ where $g(a) = 0$ .

Solution: L'Hôpital's rule gives $\frac{f(x)}{g'(x)}$ evaluated at $x = a$ .

Pattern 3: Integral Equations

Find $f$ if $\int_0^x f(t)\,dt = G(x)$ .

Solution: Differentiate both sides: $f(x) = G'(x)$ .

Pattern 4: Both Limits Variable

Find $\frac{d}{dx}\int_{\phi(x)}^{\psi(x)} f(t)\,dt$ .

Solution: $f(\psi)\psi' - f(\phi)\phi'$ .

19. Final Remarks

The Fundamental Theorem of Calculus is arguably the most important result in elementary calculus. It unifies two seemingly disparate problems—finding tangent lines (differentiation) and computing areas (integration)—revealing them as inverse operations.

This connection has profound implications throughout mathematics, physics, engineering, and beyond. From computing work and energy to modeling population growth, from signal processing to probability theory, FTC provides the foundational bridge between rates and accumulations.

As you continue your study of calculus, you will see FTC appear again and again—in integration techniques, in differential equations, in multivariable calculus (as the various generalized Stokes' theorems), and in real and complex analysis. Mastering FTC now will pay dividends throughout your mathematical journey.

Key Points to Remember

• FTC Part 1: Differentiation undoes integration
• FTC Part 2: Integration undoes differentiation (up to constant)
• Variable limits require chain rule (Leibniz rule)
• Not all integrands have elementary antiderivatives
• FTC connects local (derivative) with global (integral)
• Continuity of integrand ensures differentiability of integral function
• Practice with variable limit problems builds essential skills
• The error function erf(x) and Fresnel integrals are defined via FTC

Fundamental Theorem of Calculus

1. Variable Upper Limit Integral

2. FTC Part 1: Derivative of the Integral

3. Newton-Leibniz Formula (FTC Part 2)

4. Differentiation with Variable Limits

5. Applications and Examples

6. Common Mistakes

❌ Wrong: ddx∫abf(t) dt=f(x)\frac{d}{dx}\int_a^b f(t)\,dt = f(x)dxd​∫ab​f(t)dt=f(x)

❌ Forgetting the Chain Rule

❌ Wrong sign for lower limit

❌ Assuming all integrals have elementary antiderivatives

7. Additional Examples

8. Practice Problems

Problem Set A: Newton-Leibniz Formula

Problem Set B: Differentiation of Integrals

Problem Set C: Limits and Applications

9. Deeper Insights

Why FTC is Revolutionary

Unification

Computational Power

Existence of Antiderivatives

Physical Interpretation

10. Historical Context

Isaac Newton (1642-1727)

Gottfried Wilhelm Leibniz (1646-1716)

James Gregory (1638-1675)

Isaac Barrow (1630-1677)

11. Summary of Key Results

12. What's Next?

13. Study Tips

Memorize the Two Parts

Master the Leibniz Rule

Check Continuity

Practice L'Hôpital Applications

14. Challenge Problems

Challenge 1: Nested Integrals

Challenge 2: Integral Equation

Challenge 3: Limit with Parameter

Challenge 4: Generalized Leibniz

15. Connections to Other Topics

Differential Equations

Physics Applications

Probability Theory

Complex Analysis

16. Quick Reference Card

Key Formulas

Common Patterns

Properties of F(x)F(x)F(x)

Conditions

Common Errors

17. Extended Examples

Frequently Asked Questions

Why is FTC so important?

Does every integrable function have an antiderivative?

What if f is discontinuous?

Can the lower limit also vary?

Why does the notation F(x)|ₐᵇ mean F(b) - F(a)?

FTC Part 1

Newton-Leibniz

Leibniz Rule

Key Insight

Continuity of FFF

Existence

Decision Guide: Which Formula?

Quick Decision Guide

When to Use FTC Part 1

When to Use Newton-Leibniz

18. Common Exam Patterns

Pattern 1: Direct Differentiation

Pattern 2: Limit Problems

Pattern 3: Integral Equations

Pattern 4: Both Limits Variable

19. Final Remarks

Key Points to Remember

❌ Wrong: $\frac{d}{dx}\int_a^b f(t)\,dt = f(x)$

Properties of $F(x)$

Continuity of $F$