Integrate x/(x^2+1) from 0 to 2: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{0}^{2} \frac{x}{{x}^{2}+1}\,dx$
2. 2. Step by step
   1. $$\int_{0}^{2} \frac{x}{{x}^{2}+1}\,dx$$
      Start with the definite integral
   2. $$\int \frac{x}{{x}^{2}+1}\,dx = \frac{\ln\left(\left|{x}^{2}+1\right|\right)}{2}$$
      Logarithm rule: the numerator is a constant multiple of the denominator's derivative, ∫ u′/u dx = ln|u|
   3. $$F(x) = \frac{\ln\left(\left|{x}^{2}+1\right|\right)}{2}$$
      Antiderivative
   4. $$F(2) - F(0) = 0.8047189562 - 0 = 0.8047189562$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{0}^{2} \frac{x}{{x}^{2}+1}\,dx = 0.8047189562$$

   ∫ from 0 to 2 = 0.8047189562

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 0.8047189562 ✓

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