Definite integral of e^(2x) from 0 to 1: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{0}^{1} {e}^{2x}\,dx$
2. 2. Step by step
   1. $$\int_{0}^{1} {e}^{2x}\,dx$$
      Start with the definite integral
   2. $$\int {e}^{2x}\,dx = \frac{{e}^{2x}}{2}$$
      Exponential rule: ∫ eᵘ du = eᵘ (divide by the inner slope)
   3. $$F(x) = \frac{{e}^{2x}}{2}$$
      Antiderivative
   4. $$F(1) - F(0) = 3.694528049 - 0.5 = 3.194528049$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{0}^{1} {e}^{2x}\,dx = 3.194528049$$

   ∫ from 0 to 1 = 3.194528049

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 3.194528049 ✓

Evaluate definite integrals with the fundamental theorem of calculus.

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