Integration Flashcards
E^x
E^x
1/x
Lnx
Sin x
-cos x
Cos x
Sin x
Sec^2x
Tan x
Cosecx Cotx
Cosecx
Cosec^2x
-cotx
Secxtanx
Secx
E^f(x)
1/f’(x) e^f(x)
Cosf(x)
1/f’(x) sinf(x)
Sin f(x)
-1/f’(x) cos f(x)
Sec^2f(x)
1/f’(x) tan f(x)
Cosecf(x)cotf(x)
1/ f’(x) cosec f(x)
Cosec^2f(x)
-1/f’(x) cotf(x)
Secf(x)tanf(x)
1/f’(x) secf(x)
1/f(x)
1/f’(x) lnf(x)
1/(f(x))^n when n can’t equal 1
1/f’(x)(n+1) F(x)^-(n+1)
Tan x
Ln |secx|
Cot x
Ln |sinx|
Sec x
Ln[ secx +tan x] + c
Cosecx
-ln [ cosecx + cotx ] + c
Ln x
X ln x - x + c
A^x
A^x/ln a
How do you differentiate using u sub
Let u equal something in the integral Differentiate with respect to x Make dx the subject Substitute back in Something will cancel out Swap u back for your blue of x
What must you remember when integrating
+c
Integrating using partial fractions
Split into partial fractions
Then differentiate as a function of 1/f(x)
What happens if your numerator is bigger than the denominator
Must divide first
What do you do if you have a definite integra
U sub
Then integrate normally as if it is a u sub
Must change the limits to u values rather than x values
Then solve
Trig integration with powers of 2 for tan/cot/cosec/sec
Use tan trig identity then integrate
Trig integration with powers of 2 for sin and cos
Use the cos double angle formula
Rearrange then substitute
What happens if you have a power of 2 for sin and cos but with f(x) not just x
Then alter the double angle formula
E.g. the sin angle was 3x.
Then cos6x= 1-2sin^23x
Even powers of sin and cos
Factorise so you have your power of 2 inside the brackets then raise it to a higher power
Replace the trig^2 with the double angle formula
Expand the brackets
Solve
Odd powers of sin and cos
Take a factor of trig^2 out
Use the sin^2 x + cos^2 x = 1
Then let u equal whichever is squared
Normal u sub
How to integrate when you have both sin and cos multiplied with different values of x
Use the identities sin a+- b ad cos +- b
Re write using the values of x that you have been given in the question
Add or subtract your two values depending on what is in the equation
Create an expression for what is given in the original equation
Should have a basic trig addition or subtraction now to integrate
Use of trig identities with no substitution
If there is a 1/ square number- x^2 could be rooted or not
Use the trig identities
If there is a root then use s^2 + c^2 = 1
If there isn’t a root then use t^2 + 1 = sec^2
How to integrate by parts
1st x integral of the second - integrate the (integral of the 2nd x differential of the first )
How do you integrate parametric
Integral with d and c as limits then y x dx/dt dt
What is the trapezium rule
Half x width of trapeziums x (ends = (2x middles) )
Differential equations by direct integration
KEEP DY WHERE IT IS
Separate your variables
Integrate oth sides
Add a plus c on one side
If there is a ln in your differential equation what do you do
Make the constant in the form of ln
If it is a general differentiation then what do you do
Leave the c
If it is a particular integral the what do you do
Need to find c
Gradient
Dy/dx
Acceleration
Dv/dt
Rate
D_/dt
Loss
Negative
Proportional
K
Inversely proportional
K/ _
How do you work out a differential equation in context
Want = got x need
