Definite integral of x sin(x) from 0 to pi: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{0}^{\pi} x \sin\left( x\right)\,dx$
2. 2. Steps, part 1
   1. $$\int_{0}^{\pi} x \sin\left( x\right)\,dx$$
      Start with the definite integral
   2. $$\int u\,dv = uv - \int v\,du$$
      Integration by parts (tabular method)
   3. $$\int x \sin\left( x\right)\,dx = \sin\left( x\right)- x \cos\left( x\right)$$
      Combine the alternating products
   4. $$F(x) = \sin\left( x\right)- x \cos\left( x\right)$$
      Antiderivative
3. 3. Steps, part 2
   1. $$F(\pi) - F(0) = 3.141592654 - 0 = 3.141592654$$
      Apply the fundamental theorem of calculus
4. 4. The answer
   $$\int_{0}^{\pi} x \sin\left( x\right)\,dx = 3.141592654$$

   ∫ from 0 to π = 3.141592654 = π

5. 5. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 3.141592654 ✓

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