An Algorithm For Integration

  1. Do you know the integral without thinking:  Example--  sec2x.  if not ...

  2. Is there some algebra you can do to make it a known integral:  Example-- (1+x)/x = 1 + 1/x.

  3. Is there a substitution that will work.  Note the du must be a factor of the integrand.  A list of good u's is below:

    1. Inside the parenthesis. 

    2. Inside the square root.

    3. The exponent.

    4. The denominator.

    5. Inside the trig function.

    6. A simple function where its derivative is a factor.

    A list of bad u's is below:

    1. u = x

    2. u = 0

    3. any u that is difficult to differentiate.

    If there is no substitution, then ...

  4. Do you see a term of the form a2 - x2, a2 + x2, or x2 - a2? If so try x = asin(u), x = atan(u), or x = asec(u) respectively.  Example--  1/(1+x2)2  If not then ...

  5. Do you see an inverse function or a product of two different kind of functions.  If so try integration by parts.  int u'v = uv - int vu'.  Example--  x2 sinx.  If not then ...

  6. Is the integrand is a product of powers of sinx and cosx.  If the power of either sinx or cosx is odd (2k + 1), break up that power into (sin2kx)(sinx) [or (cos2kx)(cosx)] and use u = sinx [or u = cosx].  If the powers are both even use the formulae:

    1. sin2x  =  1/2 (1 - cos(2x))

    2. cos2x =  1/2 (1 + cos(2x))

  7. If the integrand is a product of secx and tanx then

    1. If it is sec2kx tannx, pull out a sec2x and convert the rest of the sec2k-2x into (1 + tan2x)k-1  and let u = tanx.

    2. If it is secmx tan2k+1x, pull out a secx tanx, and convert the tan2kx into (sec2x - 1)k and let u = secx.

    3. If the integrand if of the form tan2kx with k positive, convert to tan2k-2x (sec2x -1).  Then multiply out and continue again if necessary.

    4. Otherwise convert to sines and cosines.

    If this does not work then ...

  8. Try Partial Fraction Decomposition.  If this still does not work ...

  9. Be creative, get tutoring (Winter 00 Tutor Schedule), or ask your instructor