The average value of $f ( x ) = \frac { x ^ { 2 } + x + 1 } { x + 3 }$ over the interval 0 $\le$ x$\le$ 2 is $\frac { 7 \ln \left( \frac { 5 } { 3 } \right) } { 2 } - 1$ .

Question

Quizplus · Accepted Answer

The average value of a function $ f(x) $ over an interval $[a, b]$ is given by $\frac{1}{b-a} \int_{a}^{b} f(x) \, dx$. Calculating the integral of $ f(x) = \frac{x^2 + x + 1}{x + 3} $ from 0 to 2 and dividing by 2 gives the stated result.

The Average Value Of f(x)=x2+x+1x+3f ( x ) = \frac { x ^ { 2 } + x + 1 } { x + 3 }f(x)=x+3x2+x+1​

The Average Value Of $f ( x ) = \frac { x ^ { 2 } + x + 1 } { x + 3 }$