Question

Evaluate: \( \int_0^1 (2x+1) dx \)

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Remember to integrate each term of a polynomial separately. For definite integrals, always evaluate the antiderivative at the upper limit and subtract its value at the lower limit. Pay attention to constants and signs.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks to evaluate a definite integral of a polynomial function over the interval from 0 to 1.

Step 2: Key Formula or Approach:
1. Perform indefinite integration of the function.
\[ \int (ax^n + b) dx = a\frac{x^{n+1}}{n+1} + bx + C \]
2. Apply the Fundamental Theorem of Calculus for definite integrals:
\[ \int_a^b f(x) dx = F(b) - F(a) \]
Where \( F(x) \) is the antiderivative of \( f(x) \).

Step 3: Detailed Explanation:
Given integral: \( I = \int_0^1 (2x+1) dx \)
First, find the indefinite integral of \( (2x+1) \):
\[ \int (2x+1) dx = 2 \int x dx + \int 1 dx \]
\[ = 2 \left( \frac{x^{1+1}}{1+1} \right) + x + C \]
\[ = 2 \left( \frac{x^2}{2} \right) + x + C \]
\[ = x^2 + x + C \]
So, the antiderivative \( F(x) = x^2 + x \).
Now, evaluate the definite integral using the limits from 0 to 1:
\[ I = [x^2 + x]_0^1 \]
\[ I = ( (1)^2 + 1 ) - ( (0)^2 + 0 ) \]
\[ I = (1 + 1) - (0 + 0) \]
\[ I = 2 - 0 \]
\[ I = 2 \]

Step 4: Final Answer:
The value of the integral is 2.