Examples using the Fundamental Theorem of Calculus

- Finding Derivatives of Integrals

h₂(x) where h₂(x) is the upper limit of integration

G(x) = ∫ f(t) dt

h₁(x) where h₁(x) is the lower limit of integration

Take derivative using the chain rule:

dG(x)/dx = f(h₂(x)) dh₂(x))/dx - f(h₁(x)) dh₁(x))/dx eq(A)

Example 1 - First doing integration, then differentiating result

t = x² x²

G(x) = ∫ sin (t) dt = - cos (t) | = ˗ [cos (x² ) ˗ 1]

t = 0 0

dG(x)/dx = sin (x²) (2x) = 2 x sin (x²) ( Result)

Next apply eq (A)

dG(x)/dx = sin (x²) (2x) - sin (0) (0) = 2 x sin (x² )

(Note you get the same result.)

Example 2 Using the chain rule given above in eq. A

e^x

G(x) = ∫ ln(t) dt f(t) = ln (t), h₂(x) = e^x , h₁(x) = e^-x , dh₂/dx = e^x, dh₁/dx = - e^-x

e^-x

Apply dG(x)/dx = f(h₂(x)) dh₂(x))/dx - f(h₁(x)) dh₁(x))/dx

dG(x)/dx = ln (e^x) e^x - ln (e^-x ) e^-x

dG(x)/dx = x e^x - -x(e^-x ) = x e^x + xe^-x ( Result)

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