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EXAMPLE 2Finding Indefinite Integrals Using Substitution

Find:

  1. (a) 5x2dx4x31
  2. (b) exex+4dx

Solution (a) Notice that the numerator equals the derivative of the denominator, except for a constant factor. So, we try substitution. Let  u=4x31. Then du=12x2dx so 5x2dx=512du. 5x2dx4x31=512duu=512duu=512ln|u|+C=512ln|4x31|+C

(b) Here, the numerator equals the derivative of the denominator. So, we use the substitution u=ex+4. Then du=exdx. exex+4dx=1ex+4exdx=1udu=ln|u|+C=u=ex+4>0ln(ex+4)+C