Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find the given indefinite integral. $$ \int(1.5)^{(1-t)} d t $$

Step-by-Step Solution

Answer

The indefinite integral of \((1.5)^{(1-t)}\) with respect to `t` is \(\frac{(1.5)^{(1-t)}}{\ln(1.5)} + C\).

Step 1: Recognize the structure

Recognize that the function to be integrated is in the form \(a^u\), where `a` = 1.5 (a constant) and `u` = \(1-t\). The integral of such a function can be written as \( \frac{a^u}{\ln(a)}\), if `u` is a function of `t` and `a` is a constant.

Step 2: Apply the formula

Applying the formula, the indefinite integral \(\int(1.5)^{(1-t)} \, dt\) becomes \(\frac{(1.5)^{(1-t)}}{\ln(1.5)} + C\), where `C` is the constant of integration.

Step 3: Simplify

Simplify the expression, bearing in mind that \(\ln(1.5)\) is simply a constant value.

Key Concepts

Exponential Functions

Exponential functions are pivotal in calculus and often appear in diverse mathematical and real-world applications. They are functions of the form \(f(x) = a^x\), where the base \(a\) is a constant, and the exponent \(x\) is a variable. These functions have unique characteristics:

• Exponential growth: If \(a > 1\), the function grows rapidly as the value of \(x\) increases.
• Exponential decay: Conversely, if \(0 < a < 1\), the function decreases quickly as \(x\) increases.
• Always positive: Since any positive number raised to a power is positive, exponential functions never take negative values.
• Continuous and differentiable: They have derivatives that are themselves exponential functions, which is useful in calculus.

The given exercise features an exponential function with a non-trivial exponent, \((1-t)\). Understanding exponential functions' behaviors and properties lays a solid foundation for tackling related integration problems.

Integration Techniques

Integration techniques are useful tools in calculus for finding the antiderivative, or the integral, of a function. The indefinite integral of a function gives a family of functions whose derivative is the original function. Here are some essential integration techniques:

• **Basic Formulas:** Knowledge of standard integration formulas is crucial. These include power rule, exponential rule, and others.
• **Substitution Method:** Useful when dealing with compositions of functions, it simplifies an integral by swapping variables.
• **Integration by Parts:** A technique based on the product rule for differentiation; it is used for products of functions.

In the exercise, the core idea used is the formula for integrating exponential functions: \[ \int a^{u(t)} \, dt = \frac{a^{u(t)}}{\ln(a)} + C \]This formula applies when the exponent is a linear function of \(t\), as in \((1-t)\). Realizing when to automatically apply known integration rules can quickly help in finding solutions.

Calculus Problem Solving

Problem-solving in calculus requires understanding the relationships between functions, their derivatives, and integrals. It's crucial to not only apply formulas mechanically but also understand the transformations and manipulations involved.When solving calculus problems:

• **Understand the Problem:** Recognize the type of function you're dealing with. Is it polynomial, rational, exponential, or something else? This dictates the integration method.
• **Choose the Technique:** Select the most effective technique for integration. That could be a direct application of a formula, substitution, or another method, depending on the problem's structure.
• **Simplify:** After performing integration, simplify the resulting expression. This step may involve simplifying numerical constants like \(\ln(1.5)\), which remains constant and straightforward.
• **Reflect and Verify:** Ensure that by differentiating your answer, you can return to the original integrand. This confirms accuracy.

Developing strong problem-solving skills in calculus is achieved through practice and understanding the underlying mathematical concepts, ensuring students can handle a wide array of calculus challenges.