Question

calculate \\( \int_{2}^{7}\left(8 x+x^{2}\
ight) d x \\), given the following.
\\( \int_{2}^{7} x^{2} d x=\frac{335}{3} \quad \int_{7}^{8} x^{2} d x=\frac{169}{3} \quad \int_{2}^{7} x d x=\frac{45}{2} \\)
\\( \int_{2}^{7}\left(8 x+x^{2}\
ight) d x= \\)
(type an integer or a simplified fraction.)

Explanation:

Step1: Use integral property

According to the property of definite - integral \(\int_{a}^{b}(f(x)+g(x))dx=\int_{a}^{b}f(x)dx+\int_{a}^{b}g(x)dx\), we have \(\int_{2}^{7}(8x + x^{2})dx=\int_{2}^{7}8xdx+\int_{2}^{7}x^{2}dx\).
Also, according to the property \(\int_{a}^{b}kf(x)dx = k\int_{a}^{b}f(x)dx\) (\(k\) is a constant), then \(\int_{2}^{7}8xdx=8\int_{2}^{7}xdx\).

Step2: Substitute the given values

We know that \(\int_{2}^{7}xdx=\frac{45}{2}\) and \(\int_{2}^{7}x^{2}dx=\frac{335}{3}\).
Substitute these values into the expression:

$$
LATEXBLOCK0
$$

Answer:

\(\frac{875}{3}\)