Step 1: Understanding the Concept:
A definite integral represents the signed area under the curve of a function between two specified limits, \(x = a\) and \(x = b\).
According to the Second Fundamental Theorem of Calculus, if \(f(x)\) is continuous and \(F(x)\) is its antiderivative, then:
\[ \int_{a}^{b} f(x) dx = F(b) - F(a) \]
This means we first find the indefinite integral and then subtract the value at the lower limit from the value at the upper limit.
Step 2: Key Formula or Approach:
The rules we will use are:
1. \(\int x^n dx = \frac{x^{n+1}}{n+1}\)
2. \(\int k dx = kx\)
3. \(\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx\)
Step 3: Detailed Explanation:
Let's solve the integral step by step.
The integral is: \(I = \int_{0}^{1} (2x + 1) dx\)
Step 3.1: Find the antiderivative \(F(x)\):
Integrate the polynomial term by term:
\[ F(x) = \int (2x + 1) dx = \int 2x dx + \int 1 dx \]
\[ F(x) = 2 \left( \frac{x^2}{2} \right) + x \]
\[ F(x) = x^2 + x \]
Step 3.2: Evaluate at the limits:
Now we apply the limits from \(0\) to \(1\):
\[ I = [x^2 + x]_0^1 \]
Step 3.3: Substitute the Upper Limit (\(x = 1\)):
\[ F(1) = (1)^2 + 1 = 1 + 1 = 2 \]
Step 3.4: Substitute the Lower Limit (\(x = 0\)):
\[ F(0) = (0)^2 + 0 = 0 \]
Step 3.5: Final Calculation:
\[ I = F(1) - F(0) \]
\[ I = 2 - 0 = 2 \]
Step 4: Final Answer:
The value of the definite integral is \(2\).