Integrate sin(x)^2 from 0 to pi: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{0}^{\pi} {\sin\left( x\right)}^{2}\,dx$
2. 2. Step by step
   1. $$\int_{0}^{\pi} {\sin\left( x\right)}^{2}\,dx$$
      Start with the definite integral
   2. $$\int {\sin\left( x\right)}^{2}\,dx = \frac{x}{2}-\frac{\sin\left(2x\right)}{4}$$
      Power-reducing identity: sin² u = (1 − cos 2u)/2
   3. $$F(x) = \frac{x}{2}-\frac{\sin\left(2x\right)}{4}$$
      Antiderivative
   4. $$F(\pi) - F(0) = 1.570796327 - 0 = 1.570796327$$
      Apply the fundamental theorem of calculus
3. 3. The answer
   $$\int_{0}^{\pi} {\sin\left( x\right)}^{2}\,dx = 1.570796327$$

   ∫ from 0 to π = 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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