91Ó°ÊÓ

An antiderivative of a function is a sort of reverse operation of differentiation. It tells us which function gives us a particular derivative. In simpler terms, if you differentiate an antiderivative, you get the original function back.
To find the antiderivative of a given function, we need to think about which function, when differentiated, will give the function we started with. For instance, consider the function in the original exercise is given by \[ f(x) = 5 \cdot 0.36^x \].
To determine its antiderivative, recall the rule that the antiderivative of a constant raised to a variable power, like 0.36 to x, involves dividing by the natural logarithm of the base. Thus, the antiderivative becomes \[ \frac{5}{\ln(0.36)} \cdot 0.36^x + C \] where C is the constant of integration.
Finding antiderivatives is a crucial step in solving integrals, especially when dealing with improper integrals.

Infinite limits are a concept used to deal with integrals that have an infinite bound. These integrals are called improper integrals, and they occur frequently in calculus when we are dealing with situations that extend indefinitely.
In solving an improper integral like the one in our exercise, \[ \int_{5}^{\infty} 5 \cdot 0.36^x \ dx \], we replace the infinite boundary with a variable, say \( b \), and take the limit as \( b \to \infty \).
This approach transforms the improper integral into a limit problem: \[ \lim_{b \to \infty} \int_{5}^{b} 5 \cdot 0.36^x \ dx \].
By using infinite limits, we can assess whether the integral converges (approaches a finite value) or diverges (increases without bound). In our example, since \( 0.36^b \to 0 \) as \( b \to \infty \), the integral converges, simplifying our computation.

The Fundamental Theorem of Calculus is a cornerstone of mathematics that links the concepts of differentiation and integration. It essentially states that if you have a continuous function, the derivative of the integral of that function, over some interval, gives you back the function itself.
This theorem has two parts, and in the context of our exercise, we use the second part, which helps to compute a definite integral using antiderivatives.
After finding the antiderivative \( \frac{5}{\ln(0.36)} \cdot 0.36^x + C \), we apply the theorem by evaluating it at the bounds of integration: \[ \left[ \frac{5}{\ln(0.36)} \cdot 0.36^x \right]_{5}^{b} \].
This means substituting the bounds into the antiderivative and subtracting, resulting in \[ \frac{5}{\ln(0.36)} \left( 0.36^b - 0.36^5 \right) \].
The Fundamental Theorem simplifies the process of integration vastly, allowing us to compute definite integrals by working with antiderivatives.