Q.6. Given that

$$\int_{\log_2 5}^{\log_2 20} x \, dx = \log_2 M$$

What is the value of $M$?

Answer Choices

• 4
• 15
• 16
• 20
• 25
• 100
• 10,000

Solution

We start by evaluating the integral:

$$\int_{\log_2 5}^{\log_2 20} x \, dx = \frac{x^2}{2} \Bigg|_{\log_2 5}^{\log_2 20} = \frac{(\log_2 20)^2 - (\log_2 5)^2}{2}$$

Using the property $\log_2 20 = 2 + \log_2 5$, we substitute and simplify:

$$\frac{(2 + \log_2 5)^2 - (\log_2 5)^2}{2} = 2 + 2\log_2 5$$

Thus, we have:

$$\log_2 M = 2 + 2\log_2 5 = \log_2 (4 \cdot 5^2)$$

Therefore:

$$M = 100$$

The value of $M$ is 100.