When the lower limit of integration is the variable of differentiation

The form of the integral must exactly match the form in the statement of the fundamental theorem of calculus in order to directly use that theorem to find the derivative of the integral. For example, the lower limit of the integral must be constant and the upper limit must be the variable of differentiation. Fortunately integrals that are not exactly of the correct form can still be differentiated.

One variant arises if the variable of differentiation is the lower limit of integration while the upper limit is constant, for example:

Example 1: Find

The fundamental theorem of calculus requires the variable of differentiation to be the upper limit of the definite integral. The situation in Example 1 is easily handled once you remember that switching the limits of integration negates an integral:

Example 1, continued: To find the derivative of the integral, we first switch the order of the limits and then apply the fundamental theorem of calculus:

Try the following derivative yourself (roll over the expression to see the answer once you have it figured out).

Example 2: Complete:

(Note the roles of t and x have been reversed in this Example, but the theorem still applies with the same reversal. The answer is a function of t.)

More:

• Why must we use two different variables?
• Other variations
  • When one limit or the other is a function of the variable of differentiation
  • When both limits involve the variable of differentiation
  • Other tricks
• Common mistakes on derivative of integral problems
  • Common types of mistakes
  • The Common Calculus Mistakes section of mathmistakes.info includes mistakes on derivatives of integrals.