Integration

Question 1

How do you integrate?

Answer 1

Write down the coefficient and the x  (e.g. 2x)
Add 1 to the power 
Divide by the new power
Add a +c at the end
Simplify
(e.g. f 2x^3 dx = 2/4x^4 + c)

Question 2

• remember to put in integral sign and dx*

* Remember + c at end of simple integration questions*

Answer 2

;.

Question 3

When do you need to prepare before integrating?

Answer 3

x as a denominator

x being square rooted

Question 4

If you have to divide and its already a fraction, times the divisor and the denominator together

Question 5

What are you supposed to do when a question says express y in terms of x 
or
express f(x) in terms of x?

Answer 5

Find the particular solution

integrate, +c, find c, rewrite

Question 6

How do you change a differentiated equation back into the original equation? (The particular solution)

Answer 6

f(x) = integration of equation in terms of dx
Integrate - remember + c
Simplify
Use point/additional piece of information (x,y) or f(x) = y
Sub values into new equation
Solve for c
Sub c into equation

Question 7

What is a definite integral?

Answer 7

Limits on the integration signs

big f, no at top and bottom

Question 8

How do you work out definite integrals (the big f with a no at the top and the bottom)?

Answer 8

Prepare
Put into square brackets
Integrate values individually
In brackets, sub in top value (upper limit) into new equation
Put a - sign between brackets
In next bracket, sub in bottom value (lower limit) into equation
Simplify brackets 
*watch out for the negative signs*

Question 9

How do you find the area under a curve?

Answer 9

Use where the shading starts as the lower limit, and where it finishes as the upper limit
Integrate terms in square brackets
Use round brackets to sub in upper and lower limits
Solve
If there is more than one area, work them out separately then add together to find the total area
If under the x axis, the area will be negative. Just make it positive

Question 10

What do you do if it is unclear where the shading of a curve starts/stops?

Answer 10

‘Cuts x-axis when y=0’
Factorise equation
Check range available and choose correct values

Question 11

How do you work out the area between curves?

Answer 11

(Use ‘Let y=y’ and factorise to find points of intersection)
Look at which curve is on top and put that one first
(e.g. integral of (line on top) - (line on bottom) with a an upper limit of___ and a lower limit of ___ in terms of dx)
Complete like a definite integral
If the lines swap around, do another calculation and add them together