Definite integral of 1/sqrt(x) from 1 to 4: animated lesson

1. 1. The problem
   Evaluating an integral

   $\int_{1}^{4} \frac{1}{\sqrt{x}}\,dx$
2. 2. Steps, part 1
   1. $$\int_{1}^{4} \frac{1}{\sqrt{x}}\,dx$$
      Start with the definite integral
   2. $$\int {x}^{\frac{-1}{2}}\,dx = 2{x}^{\frac{1}{2}}$$
      Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)
   3. $$\int 1{x}^{\frac{-1}{2}}\,dx = 2{x}^{\frac{1}{2}}$$
      Constant multiple rule: ∫ k·f dx = k∫ f dx
   4. $$F(x) = 2{x}^{\frac{1}{2}}$$
      Antiderivative
3. 3. Steps, part 2
   1. $$F(4) - F(1) = 4 - 2 = 2$$
      Apply the fundamental theorem of calculus
4. 4. The answer
   $$\int_{1}^{4} \frac{1}{\sqrt{x}}\,dx = 2$$

   ∫ from 1 to 4 = 2

5. 5. Check
   • ✓Differentiated the antiderivative and compared it with the integrand ✓
   • ✓Adaptive Simpson's rule gives 2 ✓

Evaluate definite integrals with the fundamental theorem of calculus.

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