Differentiation under the integral sign

Differentiation under the Integral Sign:

Differentiation under the integral sign allows us to differentiate an integral that depends on a parameter. Suppose we have an integral of the form:

F(x)=a(x)b(x)f(x,t)dt

Where:

  • F(x) is a function of x.
  • f(x,t) is a continuous function of both x and t.
  • a(x) and b(x) are functions of x that define the limits of integration.

Theorem: If f(x,t) and its partial derivative fx(x,t) are continuous on some region R containing (x,t) and xa(x)b(x)f(x,t)dt exists, then:

ddx(a(x)b(x)f(x,t)dt)=ddxF(x)=a(x)b(x)xf(x,t)dt+f(x,b(x))ddxb(x)f(x,a(x))ddxa(x)

This theorem provides a way to differentiate a function that involves a varying limit of integration with respect to a parameter x by applying the derivative to both the integrand and the limits of integration.

Key points to remember:

  1. Conditions for Differentiation under the Integral Sign:

    • f(x,t) and its partial derivative fx(x,t) should be continuous.
    • The integral limits should also depend on x and be differentiable.
  2. Application:

    • Useful in problems involving changing limits of integration, parametric integrals, or differential equations involving integrals with varying parameters.
  3. Use Caution:

    • The conditions for the theorem must be checked before applying it, as not all integrals permit differentiation under the integral sign.
  4. Example:

    • Suppose we have the integral:

      F(x)=0x2extdt

      We aim to find ddxF(x) using differentiation under the integral sign.

      Solution:

      1. Given Integral: F(x)=0x2extdt

      2. Apply Differentiation under the Integral Sign:

        According to the theorem, ddx(a(x)b(x)f(x,t)dt)=a(x)b(x)xf(x,t)dt+f(x,b(x))ddxb(x)f(x,a(x))ddxa(x)

      3. Derivative Calculation:

        Let's differentiate F(x) using the theorem:

        ddxF(x)=ddx(0x2extdt)

        We differentiate with respect to x within the integral:

        x(ext)=text

        Applying the theorem:

        ddxF(x)=0x2textdt+exx22xex00

      4.  ddxF(x)=0x2textdt+2xex2

      5. Result:

        Thus, the derivative of F(x) using the Differentiation under the Integral Sign theorem is: ddxF(x)=0x2textdt+2xex2