11. Integration Flashcards
∫a(bx+d)^e
a/b X 1/(e+1) (bx+d)^e+1 + c
∫a/(bx+d)
a/b ln(|bx+d|) + c
∫a cos(bx+d)
a/b sin(bx+d) + c
∫a sin(bx+d)
-a/b cos(bx+d) + c
∫sec^2 x
tan(x) + c
∫cosec(x)cot(x)
-cosec(x) + c
∫cosec^2(x)
-cot(x) + c
∫sec(x)tan(x)
sec(x) + c
∫a((f(x))^n where a is a multiple of f’(x)
1/(n+1) X a/f’(x) X (f(x))^n+1 + c
∫af(x)/g(x) where f(x) is a multiple of g’(x)
af(x)/g’(x) ln(g(x))+ c
∫f(x) X e^g(x) where f(x) is a multiple of g’(x)
f(x)/g’(x) e^g(x) + c
Substitution for sin^2(x)
1/2 - 1/2(cos2x)
Substitution for cos^2(x)
1/2 + 1/2(cos2x)
Substitution for tan^2(x)
sec^2(x) - 1
Powers of sec and cosec
Don’t change in integration
Reverse chain rule
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
Integration by Substitution
- Rearrange for x in terms of u and dx in terms of du by finding du/dx
- Substitute in for x and dx and simplify
- Take the integral of that
- Write answer in terms of x (don’t need to expand)
Definite Integration by Substitution
Find u for each x and integrate between those
∫u(dv/dx) dx
uv - ∫v(du/dx) dx
Integration by parts process
- 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
- Find v and du/dx
- Substitute into the formula
Repeated integration by parts
When you get a minus the original integral add that to both sides and half
Integrating ln(x)
Integrate 1 ln(x) by parts
When to use partial fractions
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)
Area between two curves
Subtract the lower equation from the higher or integrate separately and subtract depending on if it is possible
Setting up differential equations
set dy/dx or other proportional to what the question says
replace with dy/dx = k…
if it is decreasing use -k, k>0
Solving differential equations
1 Use the proportion to write the derivative in terms of k
- Put all y terms on the LHS and x on the RHS with the respective d(variable)
- Integrate and add a +c to one side
- Rearrange into a nice/requested form
- Substitute for c if required
Showing what happens as t increases, or showing a maximum value for differential equations
Show what happens as t tends to infinity
Trapezium rule
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
Finding h for trapezium rule
(b-a)/n where b and a are the maximum and minimum limits and n the number of trapeziums
Parametric integration
Multiply the y equation by the differential of the x one and integrate with respect to t
Integrate between limits of t
Integration as the limit of the sum
Summing a function of x δx (lim δ->0) is the same as integrating that function between the limits
