Advanced Substitution   Substituting x   ExampleFind           SolutionNotice first that the square root is what makes this problem difficult.  Hence we let         u  =  x - 1         du  =  dx This seems not to get rid of the x term, but notice that if we add 1 to both sides, we get         x  =  u + 1 Now we can substitute to get                   =  2/5 u5/2 + 2/3 u3/2 + C           =  2/5 (x - 1)5/2 + 2/3 (x - 1)3/2 + C   Exercises:   Find the following integrals                                   Summary of Substitution A list of "good u's" follows: A.  Inside the parentheses B.  In the square root C.  The denominator E.  In the exponential F.  Be creative    Definite IntegralsWe can use substitution to evaluate definite integrals.  When using substitution, we substitute the values for u and do not resubstitute at the end. ExampleEvaluate        SolutionThis is a substitution with         u  =  1 + x3         du  =  3x2 dx        1/3 du  =  x2 dxThis is just old substitution business.  The new part is how to deal with the limits 0 and 1.  We just plug into x and find u.        When      x  =  0        u  =  1 + 03  =  1        When      x  =  1        u  =  1 + 13  =  2Now we can substitute all the numbers and expressions to get        Now we just plug in the numbers 2 and 1.  Notice that there is no " + C" since it is a definite integral.  We also do not need to resubstitute since the numbers already represent "u" values.  Plugging in gives        1/3 ( ln 2  -  ln 1)  =  1/3 ln 2   Back to the Math Department Home e-mail Questions and Suggestions