1. if $\\int_{a}^{b}f(x)dx = a + 2b$, then $\\int_{a}^{b}(f(x)+5)dx=$ (a) $a + 2b + 5$ (b) $5b - 5a$ (c) $7b…
1. if $\\int_{a}^{b}f(x)dx = a + 2b$, then $\\int_{a}^{b}(f(x)+5)dx=$ (a) $a + 2b + 5$ (b) $5b - 5a$ (c) $7b - 4a$ (d) $7b - 5a$ (e) $7b - 6a$
Answer
Explanation:
Step1: Use the integral property
$$\int_{a}^{b}(f(x)+5)dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}5dx$$
Step2: Substitute the given value
Since $\int_{a}^{b}f(x)dx = a + 2b$, and $\int_{a}^{b}5dx=5x\big|_{a}^{b}=5(b - a)=5b-5a$.
Step3: Combine the results
$$\int_{a}^{b}(f(x)+5)dx=(a + 2b)+(5b-5a)=7b-4a$$
Answer:
C. $7b - 4a$