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.