Blog #15

Antiderivatives/Integrals:

First of all, an antiderivative is a general solution, while an integral is more specific where you must solve for c.
Second, an antiderivative is just the derivative backwards. You will be given f’(x) and asked to find f(x).
Third, how to find an antiderivative: If your given ax^b (a being a constant and b being an exponent), your formula is (a/b+1)(x^b+1).
*Note: When solving for the antiderivative, always add “+ c” in place of a constant. When solving for an integral, you will be given something like “f(1) = 2”, in that case, plug in 1 for x into your antiderivative and set it equal to 2 and that’s your constant.
**Note: When given the second derivative, just solve twice.
***Note: When given an integral, it is always used with the integral symbol in front (“S”) and dx afterwards.
****Note: Shortcut: If you have a fractional exponent, add one and multiply by the reciprocal.
*****Note: Trig functions: You just do the opposite of the derivative.

Ex. 1) S dx = x + c
Ex. 2) S (x+2) dx = (1/1+1)(x^1+1) = x^2/2 + 2x + c
Ex. 3) S (3x^4 – 5x^2 + x) dx = 3x^5/5 – 5x^3/3 + x^2/2 + c
Ex. 4) S sinx dx = -cosx
Ex. 5) S x^1/2 dx = (1/2) + 1 = 3/2 = 2x^(3/2)/3 + c
Ex. 6) S 3x^2 dx = x^3 + c

Ex. 7) Find the integral of f’(x) = 1/x^2 that satisfies the initial f(1) = 0.
S x^-2 dx = (1/-1)(x^-1) = -1/x + c
-1/1 + c = 0
-1 + c = 0
c = 1
-1/x + 1