Definite integral of ln(x) from 1 to 2: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{1}^{2} \ln\left( x\right)\,dx$
2. 2. Step by step
   1. $$\int_{1}^{2} \ln\left( x\right)\,dx$$
      Start with the definite integral
   2. $$\int \ln\left( x\right)\,dx = x \left(\ln\left( x\right)-1\right)$$
      ∫ ln x dx = x ln x − x (integration by parts)
   3. $$F(x) = x \left(\ln\left( x\right)-1\right)$$
      Antiderivative
   4. $$F(2) - F(1) = -0.6137056389 - \left(-1\right) = 0.3862943611$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{1}^{2} \ln\left( x\right)\,dx = 0.3862943611$$

   ∫ from 1 to 2 = 0.3862943611

4. 4. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 0.3862943611 ✓

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