#### To determine

**To state:**

Both parts of the Fundamental Theorem of Calculus

#### Answer

If *f* is continuous on *[a, b]*, then

Part 1: If *gx=∫abf(t)dt* then *g'x=f(x)*

Part 2: *∫abf(x)dx=Fb-F(a)* where *F* is any antiderivative of *f*, that is *F'=f*

#### Explanation

**Part 1:**

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

*gx=∫abf(t)dt a≤x≤b*

is continuous on *[a, b]* and differentiable on *(a, b)*, and *g'x=f(x)*

**Part 2:**

If *f* is continuous on *[a, b]*, then

*∫abf(x)dx=Fb-F(a)*

Where *F* is any antiderivative of *f*, that is, a function *f* such that *F'=f*

**Conclusion:**

Therefore,

If *f* is continuous on *[a, b]*, then

Part 1: If *gx=∫abf(t)dt* then *g'x=f(x)*

Part 2: *∫abf(x)dx=Fb-F(a)* where *F* is any antiderivative of *f*, that is *F'=f*