5.3 The Fundamental Theorem of Calculus

Question 1

What is the purpose of u-substitution in integration?

Answer 1

u-substitution simplifies integrals by letting u=g(x)

Question 2

What is the u-substitution formula?

Answer 2

d/dx(f(g(x))=f’(g(x)) times g’(x)

Question 3

When is u-substitution useful?

Answer 3

When an integral involves a composite function, and one part is the derivative of another.

Question 4

How do you reverse a u-substitution after integration?

Answer 4

Replace u with the original expression in terms of x

Question 5

Give an example problem using u-substitution and explain how to do it.

Answer 5

For ∫xcos(x^(2))dx, use u=x^(2), du=2xdx, and the integral becomes
1/2 ∫cos(u)du.

Question 6

what is the formula for the fundamental theorum of Calculus Part 1?

Answer 6

If F(x)= the integral from a to x of f(t)dt, then F’(t)=f(t)

Question 7

what is the fundamental theory of Calculus Part 1 essentially saying?

Answer 7

Differentiating an integral function gives back the original function.

Question 8

what is the formula for the fundamental theorum of Calculus Part 2?

Answer 8

If f is continuous on [𝑎,𝑏] and F is an antiderivative of f, then the integral from a to b of f(x)dx=F(b)-F(a)