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

1. 1. The problem
   Evaluating an integral

   $\int_{-1}^{1} \frac{1}{1+{x}^{2}}\,dx$
2. 2. Step by step
   1. $$\int_{-1}^{1} \frac{1}{1+{x}^{2}}\,dx$$
      Start with the definite integral
   2. $$\int \frac{1}{1+{x}^{2}}\,dx = \tan^{-1}\left( x\right)$$
      Inverse tangent form: ∫ du/(a² + u²) = (1/a)·arctan(u/a)
   3. $$F(x) = \tan^{-1}\left( x\right)$$
      Antiderivative
   4. $$F(1) - F(-1) = 0.7853981634 - \left(-0.7853981634\right) = 1.570796327$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{-1}^{1} \frac{1}{1+{x}^{2}}\,dx = 1.570796327$$

   ∫ from -1 to 1 = 1.570796327 = π/2

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 1.570796327 ✓

Evaluate definite integrals with the fundamental theorem of calculus.

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