Integration 1 | | DJS

Integration 1

Integration is seen by many as the reverse of differentiation. For initial purposes this is true, but soon you see integration as the greater opportunity in calculus. Basically, differentiation discards some of the available information - all we are doing is finding the gradient of a function, so the order of the new (polynomial) function is one less than the original. For integration the opposite is true and we must apply other knowledge to complete the new, higher order, function.

Integration requires a new set of skills and techniques. Some of the skills are:

standard - you have learned this particular integration
substitution - the variable is replaced, a new integration is formed, hopefully more manageable.
product - the integration is viewed as a product, usually of two different sorts of function (e.g. an exponential and a trig function). the formula is applied.
direct - the parts of the integration are recognised as the result of a differentiation (probably of some complexity); the result is found ‘directly’.
by parts - where the integration is a fraction of polynomial terms, usually linear and quadratic, this can be separated into partial fractions and then the results tackled in turn.

The standard integrals you can be expected to know vary with syllabus. I list the ones I expect to be recognised, but you should be able to generate each result yourself, perhaps with a little prompting. Since I have problems reproducing the integral sign, I have used an arrow, -->, to indicate the process and added the +c on the end, if only to convince you where the next one begins...  The more extreme ones (towards the further mathematicians) are on the right.

It is much more useful to learn how these are done than simply to learn the results.

Your objective is understanding, not just learning,

anx  --> anx n-1 +c          sin (ax+b) ---> a cos (ax+b) +c                 tan ax ---> a sec² ax  +c  
sin ax ---> a cos ax         2 sin² ax  = 1 - cos2x --> x - 2 sin2x + c  
A e
nx --> An e nx +c       1/x  --> ln x + c        

   1        ---->    1   ln |ax+b|  +c                        f’(x)    ---->  ln | f(x) | +c
ax+b                a                                               f(x)

    1        ---->    1   arctan x/a +c                  1        ---->   1   ln | x + a |+c   =   1   arctanh x/a +c   
a² + x²              2a                                     a² - x²               2a     | x - a |              2a

       1          ---->    1   arcsin x/a   +c             1         ---->   1   arcsinh x/a   +c   
 √(a² - x²)                a                                    √(a²- x²)             a

BBC Bitesize 7 small  pages of pretty good stuff.
Khan Academy almost 100 pages of notes.
NRich a site worth knowing about. You’re stage 4 and 5. Search.
Math is fun a different approach, pretty pretty.

Extension for FM:       Dirac Delta function

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