Integrate 1/(x^2+1) from 1 to 2: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{1}^{2} \frac{1}{{x}^{2}+1}\,dx$
2. 2. Step by step
   1. $$\int_{1}^{2} \frac{1}{{x}^{2}+1}\,dx$$
      Start with the definite integral
   2. $$\int \frac{1}{{x}^{2}+1}\,dx = \tan^{-1}\left( x\right)$$
      Inverse tangent form: ∫ du/(a² + u²) = (1/a)·arctan(u/a)
   3. $$F(x) = \tan^{-1}\left( x\right)$$
      Antiderivative
   4. $$F(2) - F(1) = 1.107148718 - 0.7853981634 = 0.3217505544$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{1}^{2} \frac{1}{{x}^{2}+1}\,dx = 0.3217505544$$

   ∫ from 1 to 2 = 0.3217505544

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 0.3217505544 ✓

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