Problem 14 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int 3 t^{-7} d t$$

Step-by-Step Solution

Answer

The short answer for the indefinite integral of the given function is: \(\int 3t^{-7} dt = -\frac{1}{2} t^{-6} + C\)

Step 1: Identify the power rule formula

We are going to use the power rule for integration, which states that: $$\int t^n dt = \frac{t^{n+1}}{n+1} + C$$

Step 2: Apply the power rule

Now, we apply the power rule formula for the given function \(3t^{-7}\). So, in this case, the value of n is -7. $$\int 3t^{-7} dt = 3 \int t^{-7} dt$$

Step 3: Integrate and simplify

Using the formula from step 1, we have: $$3 \int t^{-7} dt = 3 \frac{t^{-7+1}}{-7+1} + C$$ Simplify the expression: $$3 \frac{t^{-6}}{-6} + C$$ $$= -\frac{1}{2} t^{-6} + C$$ Therefore, the indefinite integral of the given function is: $$\int 3t^{-7} dt = -\frac{1}{2} t^{-6} + C$$

Key Concepts

Power Rule for Integration

Understanding the power rule for integration is essential in calculus — it's a fundamental technique used for finding the indefinite integral of functions that are expressed as a power of the variable. The rule is elegantly simple: if you have an integral of the form \(\int t^n dt\), where \(n\) is a real number different from \(−1\), you can integrate it by adding 1 to the exponent and then dividing by the new exponent.

The general form of the power rule for integration is given by:\[\int t^n dt = \frac{t^{n+1}}{n+1} + C\], where \(C\) is the constant of integration, representing all the unknown constants that could be added to the function after integration.

For example, when applying it to \(3t^{-7}\), we increment the exponent from \(−7\) to \(−6\) and divide by \(−6\), resulting in \(−\frac{1}{2}t^{-6}\) plus the constant of integration. This formula only fails when the exponent \(n = -1\), in which case, the result involves a natural logarithm function.

Integration Techniques

While the power rule is straightforward, there are many other integration techniques that students of calculus may need to employ. These techniques are designed to tackle integrals that are not directly solvable by the power rule. Some common methods include:

• Substitution: Useful when an integral contains a composite function.
• Integration by parts: Handy when dealing with the product of two functions.
• Partial fraction decomposition: This is used when integrating rational functions.
• Trigonometric integration: Employed for integrals involving trigonometric functions.
• Numerical integration: When analytic solutions are difficult, numerical methods such as the Simpson's rule or the trapezoidal rule are used.

Each of these techniques has its own set of rules and applications. Mastering them enables students to tackle a wide range of problems. In our example with the power rule, we used one of the simplest techniques, demonstrating that even within the realm of basic techniques, understanding the conditions and applications is crucial for success in integration.

Calculus

Calculus is an area of mathematics that studies how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. There are two major branches: differential calculus, which concerns the rate of change of quantities, and integral calculus, which focuses on the accumulation of quantities.

The concept of integration, a core component of integral calculus, is all about addition or accumulation, and it comes in two flavors: definite and indefinite. Indefinite integrals, such as the one in our exercise, provide a general function plus a constant of integration, representing the antiderivative of the function. Definite integrals yield a numerical result and are often interpreted as the area under the curve of the function on a certain interval.

Grasping the basics of calculus, including differentiation and integration rules, not only helps solve mathematical problems but also builds a foundation for understanding complex scientific concepts, from physics to economics, and beyond.