11. Integration

Question 1

∫a(bx+d)^e

Answer 1

a/b X 1/(e+1) (bx+d)^e+1 + c

Question 2

∫a/(bx+d)

Answer 2

a/b ln(|bx+d|) + c

Question 3

∫a cos(bx+d)

Answer 3

a/b sin(bx+d) + c

Question 4

∫a sin(bx+d)

Answer 4

-a/b cos(bx+d) + c

Question 5

∫sec^2 x

Answer 5

tan(x) + c

Question 6

∫cosec(x)cot(x)

Answer 6

-cosec(x) + c

Question 7

∫cosec^2(x)

Answer 7

-cot(x) + c

Question 8

∫sec(x)tan(x)

Answer 8

sec(x) + c

Question 9

∫a((f(x))^n where a is a multiple of f’(x)

Answer 9

1/(n+1) X a/f’(x) X (f(x))^n+1 + c

Question 10

∫af(x)/g(x) where f(x) is a multiple of g’(x)

Answer 10

af(x)/g’(x) ln(g(x))+ c

Question 11

∫f(x) X e^g(x) where f(x) is a multiple of g’(x)

Answer 11

f(x)/g’(x) e^g(x) + c

Question 12

Substitution for sin^2(x)

Answer 12

1/2 - 1/2(cos2x)

Question 13

Substitution for cos^2(x)

Answer 13

1/2 + 1/2(cos2x)

Question 14

Substitution for tan^2(x)

Answer 14

sec^2(x) - 1

Question 15

Powers of sec and cosec

Answer 15

Don’t change in integration

Question 16

Reverse chain rule

Answer 16

Where one component is a multiple of the single derivative of the second component
Add one to the power of the second component and differentiate
Put a multiple at the front so it goes to what you were integrating

Question 17

Integration by Substitution

Answer 17

1. Rearrange for x in terms of u and dx in terms of du by finding du/dx
2. Substitute in for x and dx and simplify
3. Take the integral of that
4. Write answer in terms of x (don’t need to expand)

Question 18

Definite Integration by Substitution

Answer 18

Find u for each x and integrate between those

Question 19

∫u(dv/dx) dx

Answer 19

uv - ∫v(du/dx) dx

Question 20

Integration by parts process

Answer 20

1. Choose a value for u and dv/dx based on which you want to integrate and differentiate, leading to integrating the product of their results
2. Find v and du/dx
3. Substitute into the formula

Question 21

Repeated integration by parts

Answer 21

When you get a minus the original integral add that to both sides and half

Question 22

Integrating ln(x)

Answer 22

Integrate 1 ln(x) by parts

Question 23

When to use partial fractions

Answer 23

If the numerator is not related to the derivative of the denominator and the denominator can be written as the product of linear factors (check for quotient)

Question 24

Area between two curves

Answer 24

Subtract the lower equation from the higher or integrate separately and subtract depending on if it is possible

Question 25

Setting up differential equations

Answer 25

set dy/dx or other proportional to what the question says
replace with dy/dx = k…
if it is decreasing use -k, k>0

Question 26

Solving differential equations

Answer 26

1 Use the proportion to write the derivative in terms of k

2. Put all y terms on the LHS and x on the RHS with the respective d(variable)
3. Integrate and add a +c to one side
4. Rearrange into a nice/requested form
5. Substitute for c if required

Question 27

Showing what happens as t increases, or showing a maximum value for differential equations

Answer 27

Show what happens as t tends to infinity

Question 28

Trapezium rule

Answer 28

y = h/2 (y0 + 2(y1 + … + yn-1) + yn)

Where h is the x-length of each trapezium and y0 -> yn the y-coordinates for x0 -> xn

Question 29

Finding h for trapezium rule

Answer 29

(b-a)/n where b and a are the maximum and minimum limits and n the number of trapeziums

Question 30

Parametric integration

Answer 30

Multiply the y equation by the differential of the x one and integrate with respect to t
Integrate between limits of t

Question 31

Integration as the limit of the sum

Answer 31

Summing a function of x δx (lim δ->0) is the same as integrating that function between the limits