Post #19

Happy late Christmas =)

Indefinite Integration:

- All the same properties of derivatives apply

Polynomials: instead of subtracting one from the exponent, add one. Take the reciprocal of the exponent and put it in front of the variable instead of bringing exponent to front.

Don't forget +c.

Examples: integrate x^3 dx

3+1 = 4 so 1/4 x ^4 + c

You can check your answers by taking the derivative of the function

4(1/4) = 1 4-1=3 = x^3

2x^2 + 6x + 5 integrated is:

2/3 x^3 + 3x^2 + 5x + c

Other Functions:

-Work backwards from taking a derivative

integrate sin x dx:

-cos x + c

Integrate sec^2 dx:

tan x + c

Integrate 2 csc x cot x dx:

-2 csc x + c

Definite Integrals:

b S a f(x) dx = f(b) - f(a)

Definite integrals will always equal a number.

Examples:

3 S 0 x^2 dx

First step is to integrate the function: 1/3 x ^3 on [0,3]

Then plug into the formula: 1/3(3)^3 - [1/3(0)^3] = 9

2 S 0 2x^2- 3x + 2 dx

Integrate: 2/3 x^3 - 3/2 x^2 + 2x on [0,2]

Plug in: 2/3(2)^3 - 3/2(2)^2 + 2(2) - [2/3(0)^3 - 3/2(0)^2 + 2(0)] = 10/3

Average value is similar to definite integrals except the formula is 1/b-a b S a f(a)

Example: Find the average value of f(x) = x^2 on [0,5]

5 S 0 x^2 dx

1/5-0 5 S 0 x^2 dx

Integrate: 1/5 [1/3 x^3] on [0,5]

Plug in: 1/5 [1/3 (5)^3 - 1/3 (0) ^3]

1/5 [ 125/3 - 0 ] = 125/15 = 25/3