Evaluate $\int_1^3 2x\,dx$.

How we read your problem

Evaluate the definite integral of 2 x from x = 1 to x = 3.

Step-by-step solution

1. 1

   Start with the definite integral

   $$\int_{1}^{3} 2x\,dx$$
2. 2

   Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1)

   $$\int x\,dx = \frac{{x}^{2}}{2}$$
3. 3

   Constant multiple rule: ∫ k·f dx = k∫ f dx

   $$\int 2x\,dx = {x}^{2}$$
4. 4

   Antiderivative

   $$F(x) = {x}^{2}$$
5. 5

   Apply the fundamental theorem of calculus

   $$F(3) - F(1) = 9 - 1 = 8$$

Formulas used

• Power rule

  $$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C,\ n \neq -1$$

• Fundamental theorem of calculus

  $$\int_a^b f(x)\,dx = F(b) - F(a)$$

A definite integral gives the signed area between the curve and the x-axis: area above the axis counts as positive and area below counts as negative.

Verification

• Differentiated the antiderivative and compared it with the integrand ✓
• Adaptive Simpson's rule gives 8 ✓