Indefinite Integrals; Fundamental Theorem of Calculus
Let f be continuous on [a, b]. An indefinite integral (or anti-derivative) of f is a function F which is continuous on [a, b] and differentiable on (a, b) and F'(x) = f(x) for all x ∈ (a, b). By FTC (part 1), you can find an indefinite integral of f as follows: define F(x) a∫xf(t)dt for every real number x. Then
F(b) - F(a) = a∫bf(t)dt - a∫af(t)dt = a∫bf(t)dt.
Part 2 of FTC says that G(b) - G(a) = a∫b f(t)dt for any other indefinite integral G of f. This follows from the fact that any two indefinite integrals of f differ by a constant (if F' = f and G' = f, then (F - G)' = 0, and any function whose derivative is 0 is a constant).
A definite integral is a number, while an indefinite integral is a function. Always take care to think about what type of integral is being dealt with. By the FTC, you can use indefinite integrals to compute definite integrals.
Exercise 1- Use Part 1 of FTC to find the derivative of the function g(s) = 5∫s(t - t2)8 dt.
Exercise 2- Use Part 1 of FTC to find the derivative of the function h(x) = 1∫e^x ln t dt.
Exercise 3- Evaluate the integral 0∫π/4(sec t)2 dt.
Exercise 4- Evaluate the integral 1∫2(4 + u2/u3)du.
Exercise 5- If f is continuous and g and h are differentiable functions, find a formula for d/dx g(x)∫ h(x) f(t) dt.