To determine
(a)
To explain:
The meaning of the indefinite integral ∫f(x)dx
Answer
Antiderivative of f
Explanation
The notation ∫f(x)dx is traditionally used for an antiderivative of f. Thus
∫f(x)dx=F(x) means F'x=f(x)
Conclusion:
Therefore,
∫f(x)dx means an antiderivative of f
To determine
(b)
To explain:
The connection between the definite integral ∫abf(x)dx and indefinite integral ∫f(x)dx
Answer
∫abf(x)dx=∫fxdxab
Explanation
1) Concept:
By using Part 2 ofthe fundamental theorem of calculus
2) Theorem:
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
3) Calculation:
A definite integral ∫abf(x)dx is a number whereas an indefinite integral ∫f(x)dx is a function.
The connection between the definite and indefinite integral is given by part 2 of the fundamental theorem
If f is continuous on [a, b], then
∫abf(x)dx=∫fxdxab
Conclusion:
Therefore,
∫abf(x)dx=∫fxdxab