To solve the given definite integral [tex]\(\int_{6}^{4} 5x \, dx\)[/tex], we can use the fundamental theorem of calculus. Here is a detailed, step-by-step solution:
1. Identify the integrand and the limits of integration:
The integrand is [tex]\(5x\)[/tex], and the limits of integration are from [tex]\(x = 6\)[/tex] to [tex]\(x = 4\)[/tex].
2. Find the indefinite integral (antiderivative):
We need to find the antiderivative of [tex]\(5x\)[/tex]. The antiderivative is found by integrating [tex]\(5x\)[/tex] with respect to [tex]\(x\)[/tex]:
[tex]\[
\int 5x \, dx = \frac{5}{2} x^2 + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
3. Evaluate the definite integral using the limits of integration:
According to the fundamental theorem of calculus, we evaluate the antiderivative at the upper and lower limits and subtract:
[tex]\[
\int_{6}^{4} 5x \, dx = \left[ \frac{5}{2} x^2 \right]_{6}^{4}
\][/tex]
4. Compute the value at the upper limit ([tex]\(x = 4\)[/tex]):
Substituting [tex]\(x = 4\)[/tex] into the antiderivative:
[tex]\[
\frac{5}{2} (4)^2 = \frac{5}{2} \cdot 16 = 40
\][/tex]
5. Compute the value at the lower limit ([tex]\(x = 6\)[/tex]):
Substituting [tex]\(x = 6\)[/tex] into the antiderivative:
[tex]\[
\frac{5}{2} (6)^2 = \frac{5}{2} \cdot 36 = 90
\][/tex]
6. Subtract the value at the upper limit from the value at the lower limit:
[tex]\[
\int_{6}^{4} 5x \, dx = 40 - 90 = -50
\][/tex]
Since the integral is being computed from a higher limit to a lower limit (from 6 to 4), we need to reverse the order of limits to get a positive result. By reversing the limits, the integral [tex]\(\int_{6}^{4} 5x \, dx\)[/tex] becomes [tex]\(-\int_{4}^{6} 5x \, dx\)[/tex]. Therefore:
[tex]\[ \int_{6}^{4} 5x \, dx = -(-50) = 50 \][/tex]
So, the value of the definite integral [tex]\(\int_{6}^{4} 5x \, dx\)[/tex] is [tex]\(50\)[/tex].
