Sure, let's solve the integral [tex]\( \int_{2x+4}^{x+20} 1 \, dx \)[/tex] step-by-step. This integral represents the area under the curve of the constant function [tex]\( f(x) = 1 \)[/tex] between the limits [tex]\( 2x+4 \)[/tex] and [tex]\( x+20 \)[/tex].
To solve this integral, follow these steps:
1. Identify the bounds of integration:
[tex]\[ a = 2x + 4 \][/tex]
[tex]\[ b = x + 20 \][/tex]
2. Set up the integral:
[tex]\[ \int_{2x+4}^{x+20} 1 \, dx \][/tex]
3. Recognize that integrating the constant 1 is essentially finding the difference in the limits:
When you're integrating a constant, in this case [tex]\(1\)[/tex], the integral simply becomes the difference between the upper and lower bounds times the constant.
[tex]\[ \int_{a}^{b} 1 \, dx = b - a \][/tex]
4. Substitute the bounds into the expression:
[tex]\[ b = x + 20 \][/tex]
[tex]\[ a = 2x + 4 \][/tex]
5. Calculate the difference:
[tex]\[ (x + 20) - (2x + 4) \][/tex]
6. Simplify the expression:
[tex]\[ x + 20 - 2x - 4 \][/tex]
[tex]\[ -x + 16 \][/tex]
Hence, the value of the integral is:
[tex]\[ \int_{2x+4}^{x+20} 1 \, dx = 16 - x \][/tex]
So, the result of the integral is:
[tex]\[ 16 - x \][/tex]
