Integration Flashcards
What does proper integration notation look like?
integral = ∫ (y) dx
If in the form Number(x)^(power), how do you integrate?
E.g. ∫ 5x^4 dx
- Add one to the power
- Divide by the new power
- Add constant C
E.g. ∫ 5x^4 dx = x^5 + C
If in the form Number(linear bracket)^(power), how do you integrate?
E.g. ∫ (2x - 1)^2 dx
i actually don’t remember BUT
you can expand this and integrate with normal rule
If in the form Number(bracket)^(power), how do you integrate?
E.g. ∫ (6x)(3x^2 - 1)^4 dx
- Add one to the power
- Divide by the new power
- Add constant C
E.g.∫ (6x)(3x^2-1)^4 dx = (3x^2 - 1)^5/5 + C
If in the form Number(e)^(power•x), how do you integrate?
E.g. ∫ 5e^7x dx
- Divide by the derivate of power
- Add constant C
E.g. ∫ 5e^7x dx = (5e^7x)/7 + C
If in the form Number(a)^(power), how do you integrate?
E.g. ∫ 5•4^7x dx
- Divide by the derivate of power
- Divide by ln(a)
- Add constant C
E.g. ∫ 5•4^7x dx = (5•4^7x)/(7ln(4)) + C
If in the form 1/x, how do you integrate?
E.g. ∫ 1/x dx
- ln|x|
- Add constant C
E.g. ∫ 1/x dx = ln|x| + C
note absolute values when doing this
When an integration is in a fraction, what do you do?
- Whether it can be simplified into an easier form
E.g. ∫ x^3+4x^2/x dx = ∫ x^2+4x dx - Whether you can apply log rules
- Finally apply ∫ f’(x)/f(x) dx rule
If in the form (bracket)/(bracket), how do you integrate?
E.g. ∫ 7x/(9x^2 + 5) dx
- Derive the bottom function
- Place this derivation above the bottom function
- Multiply with necessary fraction
- Add constant C
E.g. ∫ 7x/(9x^2 + 5) dx = 7/18 ln|9x^2 + 5| +C
note absolute values when doing this
If in the form cosf(x), how do you integrate?
E.g. ∫ cos(5x) dx
- Divide by f’(x)
- Change cos → sin
- Add constant C
E.g. E.g. ∫ cos(5x) dx = 1/5 sin(5x) + C
If in the form sinf(x), how do you integrate?
E.g. ∫ sin(7x) dx
- Divide by f’(x)
- Change ±sin → ∓cos
- Add constant C
E.g. E.g. ∫ sin(7x) dx = -1/7 cos(7x) + C
If in the form sec^2f(x), how do you integrate?
E.g. ∫ sec^2(3x) dx
- Divide by f’(x)
- Change sec^2 → tan
- Add constant C
E.g. ∫ sec^2(3x) dx = 1/3 tan(3x) + C
Definite integrals are (?), which are found by…
Numbers (NOT area)
- Integrate formula
- Place higher x value into integrated function
- Place lower x value into integrated function
- Subtract higher x value into lower x value
Area can be found by…
- Draw a graph
- Find x intercepts between these functions
- Integrate the functions with absolute values
The 3 cases for areas are…
- All positive (over x-axis)
- All negative (under x-axis)
- Half and half (some over some under x-axes)
If it’s the 3rd case (half and half), how do you find the area?
- split the graph and integrate from where it changes sides
There are 3 functions that are difficult to integrate. What are they and how can you solve them?
- Split functions (look at 3rd case)
- y = ln f(x) (make x subject and integrate)
- 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