A LEVEL MATHS: Integration
A decreasing function is that which has a negative...
Gradient dy/dx<0
An increasing function is that which has a positive....
Gradient dy/dx>0
How would you inegrate tanx?
The denominator differentiates to form the numerator
How do you integrate ∫tan²x dx?
Trig identities ans tanx - x + c
How do you integrate ∫cot²x?
Use trig idenity: sec²θ≡ 1 + tan²θ Answer: -cotx - x +c
The volume of inter dimensional space jelly, V, in a container is decreasing over time , t, at a rate directly proportional to the square of its volume. Show this as a differential equation
dV/dt = -kV^2
The number of bacteria in a petting fish is increasing over time, t, at a rate directly proportional to the number of bacteria at the time, b. Formulate a differential equation to show this information
db/dt = kb
What does e^x integrate to?
e^x + C
What does 1/x inegrate to?
ln(|x|) + C
Write general solution to differential equation dP/dt =-0.1P
lnP =-0.1t + C
Wha does ∫f'(x)/f(x) dx inegrate to?
ln|f(x)| + C
What does cosx integrate to?
sin(x)
What does sec²x integrate to?
tanx
How would you integrate lnx²?
Do it by parts: First you bring the 2 indice down usin laws of logs and then integrate ∫2lnx dx =2xln(x) -2x + C
What does ∫1/(ax+b) dx inegrate to?
(1/a)ln|ax+b| +C
What does e^kx integrate to?
(1/k)e^x + C
What is the formula for integrating products when some parts of SOME PRODUCTS ARE MADE UP OF A FUNCTION AND ITS DERIVATIVE (FUNCTION IS RAISED TO A POWER)
∫(n+1)f'(x)[f(x)]ⁿdx= [f(x)]ⁿ⁺¹ +C
What is the formula for the reverse chain rule?
∫du/dx x f'(u) = f(u) + C
∫sinkx dx =
-1/k coskx + C
What does sinx integrate to?
-cos(x)
∫cosecxcotx dx
-cosecx + C
∫cosec²x dx
-cotx + C
What does t= when we are referring to the starting conditions for differential equations?
0
∫coskx dx =
1/k sinx + C
∫sec²kx dx =
1/k tankx + C
How would you integrate ∫lnx?
By parts
How would you integrate sin²x?
Double Angle Formula
How would you integrate ∫cos²5x dx ?
Double angle formulae Answer: 1/2(x) + 1/20(sin(10x) +C
What is the set of rules which you need to remember when using integration by parts?
I=Inverse L=log A=algebra T=Trig E=exponentials
What is the integration pathway for trigonometry?
In addition to this: the factor formulae from the addition formulae can be helpful or integration
∫secxtanx dx
secx + C (sex c tan means sex tonight)
What is the formula for integration by parts
uv -∫(du/dx)(v)
How does you integrate ∫x^n dx?
x^(n+1)/ (n+1) + C
