Question

If $\displaystyle\int_{-1}^{4} f(x)\,dx = 4$ and $\displaystyle\int_{2}^{4} [3 - f(x)]\,dx = 7$, then $\displaystyle\int_{-1}^{2} f(x)\,dx$ equals

• A. $-2$
• B. 3
• C. 5
• D. 8

Hint:

$\displaystyle\int_a^c f\,dx = \int_a^b f\,dx + \int_b^c f\,dx$. Use this to split or combine integral limits when partial values are known.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Use the additive property of definite integrals and the given information.
Step 2: Detailed Explanation:
$\displaystyle\int_2^4[3-f(x)]dx = 6-\int_2^4f(x)dx = 7 \Rightarrow \int_2^4f(x)dx = -1$.
$\displaystyle\int_{-1}^4 f(x)dx = \int_{-1}^2+\int_2^4 \Rightarrow 4 = \int_{-1}^2 f(x)dx + (-1) \Rightarrow \int_{-1}^2 f(x)dx = 5$.
Step 3: Final Answer:
$\displaystyle\int_{-1}^2 f(x)\,dx = 5$.