Integration

Question 1

What is the anti-derivatives of 4x^3?

Answer 1

x^4 + c

Question 2

What is the anti-derivatives of 6x^2?

Answer 2

2x^3 + c

Question 3

What is the anti-derivatives of 10?

Answer 3

10x + c

Question 4

What the hell does this magical Greek letter mean?
∫
∫ elongated S

Answer 4

Is the integration symbol that informally means “undo”.

So y dash becomes y

Is finding the original.

Question 5

∫x^2 + 2x ?

Answer 5

x^2+1 / 2+1 + 2x^1+1 / 1+1

Question 6

The formula for finding the anti-derivative

Answer 6

x^n+1 / n + 1 + c

Question 7

What is the difference between definite and indefinite integrals

Answer 7

In the weird elongated Greek s, there is no indefinite number on top and bottom, but there is for definite.

Question 8

what is ∫ of 12e

Answer 8

e IS A NUMBER

e IS EULAR WEIRD NUMBER

So….

the derivative of a constant is?? some number x

so

12ex

is NOT 0 otherwise it wouldn’t be there.

but if a constant is there, the anti derivative must have x.

Question 9

∫10e - 7x^5 + 3/5sqrt(x^9)

so complicated…

Answer 9

just break it down

10e = 10ex

7x^5 = 7x^6/6

3/5sqrt(x^9) = x^9/2

do the formula

x^11/2 / 11/2

divide 11/2 with 3/5 with KCF

becomes 6/55sqrt(x^9)

overall

10ex - 7x^6/6 + 6/55 sqrt(x^9) + C

boom

Question 10

∫sqrt(7x+3)dx

function within huh?

Answer 10

ez

let u = 7x+3 = 7

so du 7 = dx

divide 7 and you get

1/7

so

1/7∫u^1/2

do the usual

1/2 + 1

u^3/2 / 3/2

1/7 / 2/3 =

2/21 u^3/2 + C

2/21 (7x+3)^3/2 + C

Question 11

how about

∫x^4 sqrt(x^5 + 2)??

Answer 11

let u = x^5 + 2

becomes 5x^4

divide 5

dx 1/5 = du x^5

since there already x^4, cancel

so now is

1/4∫u^1/2

do the usual and you get

2/15 (x^5 + 2)^3/2 + C

to be quick, just do u^1/2 in the start and put 1/5

Question 13

How do you find x in a parabola

Answer 13

Whatever the expression is,

y is always 0

Eg

y = x^2 -5x -6

0 = x^2 - 5x - 6

(for this just use the quadratic formula and make sure you use + and -)

Eg 2

y = x^2 -9

0 = x^2 - 9

Make sure you have a plus and minus answer like the parabola.