Fundamental Theorem of Calculus Math Example 4

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Example 4

medium
If F(x)=∫0xet2 dtF(x) = \int_0^x e^{t^2}\,dt, find F′(x)F'(x).

Solution

  1. 1
    By FTC Part 1, ddx∫0xf(t) dt=f(x)\frac{d}{dx}\int_0^x f(t)\,dt = f(x).
  2. 2
    Here f(t)=et2f(t) = e^{t^2}, so F′(x)=ex2F'(x) = e^{x^2}.

Answer

F′(x)=ex2F'(x) = e^{x^2}
FTC Part 1 applies directly: replace tt with xx in the integrand. Note that ex2e^{x^2} has no elementary antiderivative, yet FTC still tells us its derivative immediately without computing anything.

About Fundamental Theorem of Calculus

The theorem stating that differentiation and integration are inverse operations, linking antiderivatives to definite integrals.

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More Fundamental Theorem of Calculus Examples

Example 1 easy

Let [formula]. Find [formula] using FTC Part 1.

Example 2 hard

Let [formula]. Find [formula].

Example 3 medium

Use FTC Part 2 to evaluate [formula].

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DerivativeIntegral