Integrate x^2 + 1 from 1 to 2: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{1}^{2} {x}^{2}+1\,dx$
2. 2. Steps, part 1
   1. $$\int_{1}^{2} {x}^{2}+1\,dx$$
      Start with the definite integral
   2. $$\int {x}^{2}\,dx = \frac{{x}^{3}}{3}$$
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
   3. $$\int 1\,dx = x$$
      Constant rule: ∫ k dx = kx
   4. $$\int {x}^{2}+1\,dx = \frac{{x}^{3}}{3}+ x$$
      Sum rule: integrate term by term
3. 3. Steps, part 2
   1. $$F(x) = \frac{{x}^{3}}{3}+ x$$
      Antiderivative
   2. $$F(2) - F(1) = 4.666666667 - 1.333333333 = \frac{10}{3} \approx 3.3333333$$
      Apply the fundamental theorem of calculus
4. 4. The answer
   $$\int_{1}^{2} {x}^{2}+1\,dx = \frac{10}{3} \approx 3.3333333$$

   ∫ from 1 to 2 = 10/3 ≈ 3.3333333

5. 5. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 3.333333333 ✓

Evaluate definite integrals with the fundamental theorem of calculus.

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