Make up #10

Indefinite Integration

Integration, the basis of all Calculus classes, is probably the hardest thing ever to learn. Integration, in lame man's terms, is just the opposite of differentiation. That means that instead of going from x to 1, you from x to (1/2)x^2.

First of all, there is a new sign: its an S-shaped symbol that just stands for opposite of derivative, or integral.

Next, there are two types of integrals: Definite and Indefinite.

Indefinite Integrals are just doing the opposite of a derivative.

int x dx (1/2)x^2 +c

int 5 dx 5x +c

int cosx dx sinx +c

See the pattern there. ALWAYS ADD A +c AT THE END OF ALL INDEFINITE INTEGRALS!!!

Definite integrals just involve one more step after this, plugging in an x value or values

int x dx [2,4] (1/2)x^2 (1/2)(4^2)=8 (1/2)(2^2)=2

int x dx [2,4]=6

You have to subtract f(a) from f(b). This gives you the area under the curve of the function on the interval. Integration problems are very very common on the AP exam and all Calculus students will know everything about basic integration before they can pass the exam or even the class.

Substitution is the only trick that integration needs. Substitution takes the place of the chain rule, multiplication rule, and even the quotient rule. The steps to substitute are as follows:

1. Find derivative inside integral.
2. Substitute u for the non-derivative then differentiate u
3. plug back in with original.

Example:

int (x^2+1)(2x) dx

you may think this is impossible but its not using substitution

1. Find derivative inside integral. u=(x^2+1) du=(2x)

2. Integrate u. (1/2)u^2 +c

3. Plug back in (1/2)(x^2+1)^2 +c

You can simplify from there but I prefer not to since I'm lazy.