Learning Outcomes

Upon completing this lesson, you will be able to:

• State both parts of the Fundamental Theorem of Calculus.
• Understand the conceptual link between differentiation and integration as inverse processes.
• Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals efficiently.
• Recognize when to use the technique of integration by substitution.
• Apply the method of u-substitution to find both indefinite and definite integrals of composite functions.

Topics Covered

• The Bridge Between Differentiation and Integration
• The Fundamental Theorem of Calculus, Part 1 (FTC1)
• The Fundamental Theorem of Calculus, Part 2 (FTC2)
• Integration by Substitution

I. The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is arguably the most important theorem in all of calculus. It establishes a profound and beautiful connection between the two main branches of calculus: differentiation and integration. It shows that they are, in essence, inverse processes of each other. The theorem is split into two parts.

A. The Bridge Between Differentiation and Integration

Before the FTC, finding the exact area under a curve (a definite integral) required the difficult process of taking the limit of an infinite sum of rectangles (a Riemann Sum). Finding an antiderivative (an indefinite integral) was a completely separate problem. The FTC provides a bridge between these two seemingly unrelated concepts, allowing us to use antiderivatives to easily calculate exact areas.

B. The Fundamental Theorem of Calculus, Part 1 (FTC1)

The first part of the theorem shows how to differentiate an "area function." If we define a function $g(x)$ as the integral of another function $f(t)$ from a constant $a$ to a variable $x$, then the derivative of this new function $g(x)$ is simply the original function $f(x)$.

FTC1 Statement:

If $f$ is continuous on $[a, b]$, then the function $g$ defined by

$$ g(x) = \int_{a}^{x} f(t) ,dt $$

is continuous on $[a, b]$ and differentiable on $(a, b)$, and its derivative is:

$$ g'(x) = f(x) $$

In simpler terms, if you integrate a function and then differentiate the result, you get back to the original function.

Example 1

Find the derivative of

$$ g(x) = \int_{2}^{x} \cos(t) ,dt $$

According to FTC1, the derivative of this integral function is simply the integrand, $\cos(t)$, with the variable $t$ replaced by $x$.

Therefore,

$$ g'(x) = \cos(x) $$