To evaluate the integral [tex]\(\int_{-6}^{10} e^{-0.05 x} \, dx\)[/tex], we follow several steps to find the solution analytically. Here's a step-by-step guide:
1. Identify the Integral:
We need to evaluate the definite integral:
[tex]\[
\int_{-6}^{10} e^{-0.05 x} \, dx
\][/tex]
2. Find the Indefinite Integral:
We start by finding the indefinite integral of the function [tex]\(e^{-0.05 x}\)[/tex].
Let [tex]\( u = -0.05 x \)[/tex]. Then, [tex]\( du = -0.05 dx \)[/tex], or equivalently [tex]\( dx = \frac{du}{-0.05} = -20 du \)[/tex].
Substituting into the integrand, we get:
[tex]\[
\int e^{-0.05 x} \, dx = \int e^{u} \cdot \frac{du}{-0.05} = -20 \int e^{u} \, du
\][/tex]
The integral of [tex]\(e^{u}\)[/tex] is [tex]\(e^{u}\)[/tex]. Therefore:
[tex]\[
-20 \int e^{u} \, du = -20 e^{u} + C
\][/tex]
Converting back to the original variable [tex]\(x\)[/tex]:
[tex]\[
-20 e^{-0.05 x} + C
\][/tex]
3. Evaluate the Definite Integral:
Next, we use the indefinite integral to find the definite integral. We need to evaluate this at the given limits [tex]\(x = -6\)[/tex] and [tex]\(x = 10\)[/tex]:
[tex]\[
\left[ -20 e^{-0.05 x} \right]_{-6}^{10}
\][/tex]
This means we compute:
[tex]\[
-20 e^{-0.05 \cdot 10} - (-20 e^{-0.05 \cdot (-6)})
\][/tex]
Simplifying the exponentiations, we get:
[tex]\[
-20 e^{-0.5} - (-20 e^{0.3})
\][/tex]
This can be further simplified to:
[tex]\[
-20 e^{-0.5} + 20 e^{0.3}
\][/tex]
4. Calculate the Numerical Value:
We approximate the values of the exponentials and multiply:
[tex]\[
-20 \cdot e^{-0.5} + 20 \cdot e^{0.3} \approx -20 \cdot 0.6065 + 20 \cdot 1.3499
\][/tex]
Combining these terms:
[tex]\[
-20 \cdot 0.6065 + 20 \cdot 1.3499 \approx -12.13 + 26.998 = 14.868
\][/tex]
Finally, the evaluated integral is approximately:
[tex]\[
\boxed{14.866562957267394}
\][/tex]
