Question

calculate \\( \int_{1}^{8} 2 x d x \\), given the following.
\\( \int_{1}^{8} x d x=31.5 \\) \\( \int_{1}^{7} x^{2} d x=114 \\) \\( \int_{7}^{8} x^{2} d x=\frac{169}{3} \\)
\\( \int_{1}^{8} 2 x d x=\square \\)

Explanation:

Step1: Use the property of definite integrals

According to the property \(\int_{a}^{b}kf(x)dx = k\int_{a}^{b}f(x)dx\) (\(k\) is a constant). For the integral \(\int_{1}^{8}2x dx\), we can rewrite it as \(2\int_{1}^{8}x dx\).

Step2: Substitute the given value

We know that \(\int_{1}^{8}x dx = 31.5\). Substitute this value into \(2\int_{1}^{8}x dx\), then \(2\times31.5\).

Answer:

\(63\)