11. further integration techniques

11.1 reversing standard derivatives

  • integrating ( )n is 1/n+1 ( )n+1 but you also need to divide by the coefficient of x

    this is ONLY when (ax + b)

  • think of it like differentiating ax + b (which is a) then dividing the standard integrating process by this value ‘a’.

    integral of f(ax+b) dx = 1/a (integral of f(ax+b)) + c

11.2 integration by substitution

  1. choose your sub and = u (something inside the function, something ‘messy’),

  2. differentiate by u, then rearrange for something linking back to the original equation e.g., dx

  3. rewrite the integral in terms on ‘u’ and the result of step 2,

  4. integrate with respect to ‘u’,

  5. replace u with your original expression in terms of x,

  6. (OPTIONAL DEFINITE INTEGRALS ONLY) substitute the x-values into the u equation to change the limits. ← this way you don’t need to do 5 and substitute back.

  • (FORMULA BOOK) integral of f’(x) / f(x) dx = ln | f(x) | + c

11.3 integration by parts

  1. choose u and dv/dx

    1. use LIATE to pick u (log, inverse trig, algebra, trig, indices)

    2. then let the rest be dv/dx

  2. differentiate u (=du/dx) and integrate dv/dx (=v)

  3. (FORMULA BOOK) integral of u dv/dx dx = u v - integral of v du/dx dx

  4. simply integrate the rest and you’re DONE!

11.4 using trigonometric identities in integration

  • integrate sin²x use cos2x = 1 - 2sin² x

  • to integrate cos²x use sin2x = 2cos²x - 1

  • then, you rearrange to make it the subject and continue integrating.

  • see no link between left = right? make everything sin and cos

11.5 integrating rational fractions

  • partial fractions then integrate

  • always factorise the denominator first!