Integrate xe^(x) from -1 to 3: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{-1}^{3} x {e}^{x}\,dx$
2. 2. Steps, part 1
   1. $$\int_{-1}^{3} x {e}^{x}\,dx$$
      Start with the definite integral
   2. $$\int u\,dv = uv - \int v\,du$$
      Integration by parts (tabular method)
   3. $$\int x {e}^{x}\,dx = {e}^{x} \left( x-1\right)$$
      Combine the alternating products
   4. $$F(x) = {e}^{x} \left( x-1\right)$$
      Antiderivative
3. 3. Steps, part 2
   1. $$F(3) - F(-1) = 40.17107385 - \left(-0.7357588823\right) = 40.90683273$$
      Apply the fundamental theorem of calculus
4. 4. The answer
   $$\int_{-1}^{3} x {e}^{x}\,dx = 40.90683273$$

   ∫ from -1 to 3 = 40.90683273

5. 5. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 40.90683273 ✓

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