Question

calculate $\int_{2}^{7}3x dx$, given the following. $\int_{2}^{7}x dx = 22.5$ $\int_{2}^{7}3x dx = \square$

Explanation:

Step1: Use the property of definite integrals

According to the property of definite integrals \(\int_{a}^{b}kf(x)dx = k\int_{a}^{b}f(x)dx\) (\(k\) is a constant). Here \(k = 3\), \(a = 2\), \(b=7\) and \(f(x)=x\). So \(\int_{2}^{7}3x dx=3\int_{2}^{7}x dx\).

Step2: Substitute the given value

We are given that \(\int_{2}^{7}x dx = 22.5\). Substitute this value into the expression from Step 1. Then \(3\int_{2}^{7}x dx=3\times22.5\).

Step3: Calculate the result

\(3\times22.5 = 67.5\).

Answer:

\(67.5\)