Integration

Question 1

What does proper integration notation look like?

Answer 1

integral = ∫ (y) dx

Question 2

If in the form Number(x)^(power), how do you integrate?

E.g. ∫ 5x^4 dx

Answer 2

1. Add one to the power
2. Divide by the new power
3. Add constant C
   E.g. ∫ 5x^4 dx = x^5 + C

Question 3

If in the form Number(linear bracket)^(power), how do you integrate?
E.g. ∫ (2x - 1)^2 dx

Answer 3

i actually don’t remember BUT

you can expand this and integrate with normal rule

Question 4

If in the form Number(bracket)^(power), how do you integrate?
E.g. ∫ (6x)(3x^2 - 1)^4 dx

Answer 4

1. Add one to the power
2. Divide by the new power
3. Add constant C
   E.g.∫ (6x)(3x^2-1)^4 dx = (3x^2 - 1)^5/5 + C

Question 5

If in the form Number(e)^(power•x), how do you integrate?

E.g. ∫ 5e^7x dx

Answer 5

1. Divide by the derivate of power
2. Add constant C
   E.g. ∫ 5e^7x dx = (5e^7x)/7 + C

Question 6

If in the form Number(a)^(power), how do you integrate?

E.g. ∫ 5•4^7x dx

Answer 6

1. Divide by the derivate of power
2. Divide by ln(a)
3. Add constant C
   E.g. ∫ 5•4^7x dx = (5•4^7x)/(7ln(4)) + C

Question 7

If in the form 1/x, how do you integrate?

E.g. ∫ 1/x dx

Answer 7

1. ln|x|
2. Add constant C
   E.g. ∫ 1/x dx = ln|x| + C

note absolute values when doing this

Question 8

When an integration is in a fraction, what do you do?

Answer 8

1. Whether it can be simplified into an easier form
   E.g. ∫ x^3+4x^2/x dx = ∫ x^2+4x dx
2. Whether you can apply log rules
3. Finally apply ∫ f’(x)/f(x) dx rule

Question 9

If in the form (bracket)/(bracket), how do you integrate?

E.g. ∫ 7x/(9x^2 + 5) dx

Answer 9

1. Derive the bottom function
2. Place this derivation above the bottom function
3. Multiply with necessary fraction
4. Add constant C
   E.g. ∫ 7x/(9x^2 + 5) dx = 7/18 ln|9x^2 + 5| +C

note absolute values when doing this

Question 10

If in the form cosf(x), how do you integrate?

E.g. ∫ cos(5x) dx

Answer 10

1. Divide by f’(x)
2. Change cos → sin
3. Add constant C
   E.g. E.g. ∫ cos(5x) dx = 1/5 sin(5x) + C

Question 11

If in the form sinf(x), how do you integrate?

E.g. ∫ sin(7x) dx

Answer 11

1. Divide by f’(x)
2. Change ±sin → ∓cos
3. Add constant C
   E.g. E.g. ∫ sin(7x) dx = -1/7 cos(7x) + C

Question 12

If in the form sec^2f(x), how do you integrate?

E.g. ∫ sec^2(3x) dx

Answer 12

1. Divide by f’(x)
2. Change sec^2 → tan
3. Add constant C
   E.g. ∫ sec^2(3x) dx = 1/3 tan(3x) + C

Question 13

Definite integrals are (?), which are found by…

Answer 13

Numbers (NOT area)

1. Integrate formula
2. Place higher x value into integrated function
3. Place lower x value into integrated function
4. Subtract higher x value into lower x value

Question 14

Area can be found by…

Answer 14

1. Draw a graph
2. Find x intercepts between these functions
3. Integrate the functions with absolute values

Question 15

The 3 cases for areas are…

Answer 15

1. All positive (over x-axis)
2. All negative (under x-axis)
3. Half and half (some over some under x-axes)

Question 16

If it’s the 3rd case (half and half), how do you find the area?

Answer 16

1. split the graph and integrate from where it changes sides

Question 17

There are 3 functions that are difficult to integrate. What are they and how can you solve them?

Answer 17

1. Split functions (look at 3rd case)
2. y = ln f(x) (make x subject and integrate)
3. not bounded to an axis (subtract top function to bottom and integrate)

note for the last option, you don’t need to split the graph, i.e. whether it’s positive or negative relative to an axis is irrelevant