Take an Antiderivative by Substitution

Use Integration by Substitution to Undo:
      · Taking the Derivative of a Composition of Functions,
      · The Chain Rule for Taking Derivatives.

      Integration by Substitution is a procedure that at first may look too long to complete, but, with a little exposure, much of the time, may be completed mentally.

      The integrand must have these as "factors"

  • dx, the differential of x
  • an "outer function" evaluated at an "inner function,"
  • the derivative of the "inner function," and
  • may be off by a constant -- need a constant or have an extra constant factor.

      The strategy is to:

  • rewrite the integral using a new independent variable, u, to replace the more complicated original independent variable, x,
  • then complete the easier integration,
  • then replace the u with the x if needed.

      On this page computation is shown in two areas, the intergal equations and the computation with u.

To Take An Antiderivative By Substitution

  1. Declare the "inside function" to be u.
  2. Take the derivative of u.
  3. Multiply by dx, the differential of x to isolate du, the differential of u.
  4. Isolate the dx and any factors containing x on one side of the equation so it is (possible x factors)(dx).
  5. If needed, multiply or divide both sides by a constant so (possible x factors)(dx) is alone -- isolated -- on one side.
  6. Go back to the original integral.
  7. Put u in place of the original function.
  8. Remove the (possible x factors)(dx) expression, and replace it with the its equivalent expression which includes du.
  9. If the problem is a definite integral, rewrite the bounds of integration in terms of u, not x.
          the new upper bound b = u(b)
          the new lower bound a = u(a)
  10. Integrate the rewritten integral.
  11. If the problem is an indefinite integral, rewrite the result in terms of x, and add the + C.

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