SCool Revision Summary
SCool Revision Summary
Integration is the inverse of differentiation
The basic rules are:
Differentiation 
Integration 

x^{n} 
nx^{n1} 
x^{n} 

e^{x} 
e^{x} 
e^{x} 
e^{x} 
ln x 

sin x 
cos x 
cos x 
sin x 
cos x 
sin x 
sin x 
cos x 
tan x 
sec^{2 }x 
sec^{2 }x 
tan x 
Indefinite integrals have no limits and therefore the answer includes a '+ c'.
Definite integrals include limits and give a numerical value as an answer.
This value corresponds to the area under the graph.
Using Integration to find an area
To find the area between the yaxis and the curve rearrange the equation as x = f(y) and use area =
Integration using Substitution
When using a substitution rewrite every part of the integral in terms of 'u', the new variable. This includes the function, the dx, and the limits.
Integration by substitution is the reverse of the chain rule and therefore can be done by inspection. Fast rules are:
=
=
=
=
=
When integrating cos^{n}x and sin^{n}x, use the substitutions cos^{2}x = 1 sin^{2}x (or sin^{2}x = 1 cos^{2}x), when n is odd, and use the substitutions sin^{2}x = and , when n is even.
Integration by Parts
Integration by parts is the inverse of the product rule.
When integrating by parts using ln x, let u = ln x.
When integrating by parts using sin x, or cos x, use parts twice to get an answer in terms of the question.
Differential Equations
A differential equation is where we have to find the original equation of a function from its gradient.
If the differential equation is in terms of x we can use normal integration.
If the differential equation is in terms of x and y we use the method of separable variables.

Collect the x's on one side and the y's and the on the other side.

Integrate both sides with respect to x.

Simplify the resulting equation.
Solving a differential equation results in a general solution (with a '+ c'). An illustration of this is called a family of curves.
If we know one point on the graph we can find the particular solution.