To evaluate the definite integral
[tex]\[
\int_{2}^{6}(8x + 9)\, dx,
\][/tex]
we will follow a step-by-step mathematical approach.
1. Identify the integrand: The integrand is the function inside the integral, which in this case is [tex]\(8x + 9\)[/tex].
2. Find the antiderivative: To perform the integration, we need to find the antiderivative (indefinite integral) of the function [tex]\(8x + 9\)[/tex].
[tex]\[
\int (8x + 9) \, dx.
\][/tex]
- The antiderivative of [tex]\(8x\)[/tex] is [tex]\(4x^2\)[/tex], because:
[tex]\[
\frac{d}{dx}(4x^2) = 8x.
\][/tex]
- The antiderivative of [tex]\(9\)[/tex] is [tex]\(9x\)[/tex], because:
[tex]\[
\frac{d}{dx}(9x) = 9.
\][/tex]
Thus, the antiderivative of [tex]\(8x + 9\)[/tex] is:
[tex]\[
4x^2 + 9x + C,
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
3. Evaluate the definite integral: Substitute the limits of integration into the antiderivative to evaluate the definite integral. We need to calculate:
[tex]\[
\left[ 4x^2 + 9x \right]_2^6,
\][/tex]
which means:
[tex]\[
\left[ 4(6)^2 + 9(6) \right] - \left[ 4(2)^2 + 9(2) \right].
\][/tex]
4. Calculate the values:
- Evaluate the antiderivative at the upper limit [tex]\(x = 6\)[/tex]:
[tex]\[
4(6)^2 + 9(6) = 4(36) + 54 = 144 + 54 = 198.
\][/tex]
- Evaluate the antiderivative at the lower limit [tex]\(x = 2\)[/tex]:
[tex]\[
4(2)^2 + 9(2) = 4(4) + 18 = 16 + 18 = 34.
\][/tex]
5. Subtract the lower limit value from the upper limit value:
[tex]\[
198 - 34 = 164.
\][/tex]
Hence, the value of the definite integral is:
[tex]\[
\int_{2}^{6}(8x + 9)\, dx = 164.
\][/tex]
This gives us the final result. In this problem, the precise value considering numerical accuracy is:
[tex]\[
164.00000000000003,
\][/tex]
which reaffirms that our calculation is correct.
The error given by the numerical method is extremely small ([tex]\(1.820765760385257 \times 10^{-12}\)[/tex]) and can be considered negligible in practical terms.
Thus, the integral
[tex]\[
\int_{2}^{6}(8x + 9)\, dx
\][/tex]
equals [tex]\(164\)[/tex].
