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## Calculus, all content (2017 edition)

### Course: Calculus, all content (2017 edition)>Unit 4

Lesson 15: Fundamental theorem of calculus: chain rule

# Fundamental theorem of calculus review

Review your knowledge of the fundamental theorem of calculus and use it to solve problems.

## What is the fundamental theorem of calculus?

The theorem has two versions.

### a) $\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt=f(x)$start fraction, d, divided by, d, x, end fraction, integral, start subscript, a, end subscript, start superscript, x, end superscript, f, left parenthesis, t, right parenthesis, d, t, equals, f, left parenthesis, x, right parenthesis

We start with a continuous function f and we define a new function for the area under the curve y, equals, f, left parenthesis, t, right parenthesis:
F, left parenthesis, x, right parenthesis, equals, integral, start subscript, a, end subscript, start superscript, x, end superscript, f, left parenthesis, t, right parenthesis, d, t
What this version of the theorem says is that the derivative of F is f. In other words, F is an antiderivative of f. Thus, the theorem relates differential and integral calculus, and tells us how we can find the area under a curve using antidifferentiation.

### b) $\displaystyle\int_a^b\!\! f(x)dx=F(b)\!-\!\!F(a)$integral, start subscript, a, end subscript, start superscript, b, end superscript, f, left parenthesis, x, right parenthesis, d, x, equals, F, left parenthesis, b, right parenthesis, minus, F, left parenthesis, a, right parenthesis

This version gives more direct instructions to finding the area under the curve y, equals, f, left parenthesis, x, right parenthesis between x, equals, a and x, equals, b. Simply find an antiderivative F and take F, left parenthesis, b, right parenthesis, minus, F, left parenthesis, a, right parenthesis.

## Practice set 1: Applying the theorem

Problem 1.1
• Current
g, left parenthesis, x, right parenthesis, equals, integral, start subscript, 1, end subscript, start superscript, x, end superscript, square root of, 2, t, plus, 7, end square root, d, t
g, prime, left parenthesis, 9, right parenthesis, equals

Want to try more problems like this? Check out this exercise.

## Practice set 2: Applying the theorem with chain rule

We can use the theorem in more hairy situations. Let's find, for example, the expression for start fraction, d, divided by, d, x, end fraction, integral, start subscript, 0, end subscript, start superscript, x, cubed, end superscript, sine, left parenthesis, t, right parenthesis, d, t. Note that the interval is between 0 and x, cubed, not x.
To help us, we define F, left parenthesis, x, right parenthesis, equals, integral, start subscript, 0, end subscript, start superscript, x, end superscript, sine, left parenthesis, t, right parenthesis, d, t. According to the fundamental theorem of calculus, F, prime, left parenthesis, x, right parenthesis, equals, sine, left parenthesis, x, right parenthesis.
It follows from our definition that integral, start subscript, 0, end subscript, start superscript, x, cubed, end superscript, sine, left parenthesis, t, right parenthesis, d, t is F, left parenthesis, x, cubed, right parenthesis, which means that start fraction, d, divided by, d, x, end fraction, integral, start subscript, 0, end subscript, start superscript, x, cubed, end superscript, sine, left parenthesis, t, right parenthesis, d, t is start fraction, d, divided by, d, x, end fraction, F, left parenthesis, x, cubed, right parenthesis. Now we can use the chain rule:
\begin{aligned} &\phantom{=}\dfrac{d}{dx}\displaystyle \int_{0}^{x^3}\sin(t) \, dt \\\\ &=\dfrac{d}{dx}F(x^3) \\\\ &=F'(x^3)\cdot\dfrac{d}{dx}(x^3) \\\\ &=\sin(x^3)\cdot 3x^2 \end{aligned}
Problem 2.1
• Current
F, left parenthesis, x, right parenthesis, equals, integral, start subscript, 0, end subscript, start superscript, x, start superscript, 4, end superscript, end superscript, cosine, left parenthesis, t, right parenthesis, d, t
F, prime, left parenthesis, x, right parenthesis, equals

Want to try more problems like this? Check out this exercise.