Problem 88 - Applied Calculus

Question

The substitution method can be used to find integrals that do not fit our formulas. For example, observe how we find the following integral using the substitution \(u=x+4\) which implies that \(x=u-4\) and so \(d x=d u\). $$ \begin{aligned} \int(x-2)(x+4)^{8} d x &=\int(u-4-2) u^{8} d u \\ &=\int(u-6) u^{8} d u \\ &=\int\left(u^{9}-6 u^{8}\right) d u \\ &=\frac{1}{10} u^{10}-\frac{2}{3} u^{9}+C \\ &=\frac{1}{10}(x+4)^{10}-\frac{2}{3}(x+4)^{9}+C \end{aligned} $$ It is often best to choose \(u\) to be the quantity that is raised to a power. The following integrals may be found as explained on the left (as well as by the methods of Section 6.1). $$ \int x(x+4)^{7} d x $$

Step-by-Step Solution

Answer

\( \frac{1}{9}(x+4)^9 - \frac{1}{2}(x+4)^8 + C \).

Step 1: Choose the Substitution

Choose the substitution to simplify the integral. We set \( u = x + 4 \). This choice makes the term \((x+4)^7\) simpler to integrate.

Step 2: Express x and dx in terms of u

Since \( u = x + 4 \), we have \( x = u - 4 \). The differential is \( dx = du \) since when we differentiate \( u = x + 4 \), we get \( du = dx \).

Step 3: Substitute into the Integral

Substitute \( x = u - 4 \) and \( dx = du \) into the integral, leading to: \( \int x(x+4)^7 \, dx = \int (u - 4) u^7 \, du \).

Step 4: Simplify the Integral

Distribute the terms inside the integral: \( \int (u - 4) u^7 \, du = \int (u^8 - 4u^7) \, du \). This results in separating the integral into two simpler integrals: \( \int u^8 \, du - 4 \int u^7 \, du \).

Step 5: Integrate Each Term

Integrate each term separately: \( \int u^8 \, du = \frac{1}{9} u^9 \) and \( \int u^7 \, du = \frac{1}{8} u^8 \). Thus, the integral becomes \( \frac{1}{9} u^9 - 4 \times \frac{1}{8} u^8 \).

Step 6: Simplify the Result

Combine the expressions from the integration: \( \frac{1}{9} u^9 - \frac{1}{2} u^8 + C \), where \( C \) is the integration constant.

Step 7: Back-Substitute for x

Replace \( u \) with \( x + 4 \): \( \frac{1}{9} (x+4)^9 - \frac{1}{2} (x+4)^8 + C \). This is the antiderivative expressed in terms of \( x \).

Key Concepts

Definite Integrals

When learning about definite integrals, it's important to understand that these are just a specific type of integral which is evaluated over a given interval. In essence, a definite integral calculates the area under the curve of a function between two points, say from \(a\) to \(b\). This is in contrast to indefinite integrals, which do not have limits of integration and thus represent a family of functions, plus a constant of integration, symbolized as \(C\).

Definite integrals can be thought of as the summation of an infinite number of infinitesimally thin rectangles under the curve of the function. The formal expression for definite integrals can be written as:
\[\int_{a}^{b} f(x) \, dx\] This is where \(f(x)\) is the function being integrated, and \(a\) and \(b\) are the limits of integration. The definite integral gives us a concrete numerical value, unlike the indefinite integral, which remains symbolic unless specific limits are applied. Integrals, both definite and indefinite, are central to calculus, as they help in determining accumulated quantities, like areas, volumes, and other similar metrics.

Integrals

Integrals are fundamental to calculus and are often described as the reverse process of differentiation. While differentiation deals with breaking down a function into its rate of change, integration is about building something up, like calculating areas and totals.

There are generally two types of integrals: definite and indefinite. Indefinite integrals represent a family of functions and are written without limits, often resulting in expressions that include the constant \(C\). This constant is essential as it accounts for all the possible vertical shifts of a function. On the other hand, as explained earlier, definite integrals compute the net area under curves, providing a fixed value.

When solving integrals, techniques such as substitution can greatly simplify the process. Substitution involves changing variables to rewrite the integral in a simpler form, often making integration possible when direct methods are cumbersome. This tactic was evident in the given exercise, where the substitution \(u = x + 4\) allowed us to turn a complex expression into something more manageable.

Calculus

Calculus is a branch of mathematics that studies continuous change. It encompasses two major areas: differential calculus and integral calculus. Differential calculus focuses on finding the derivative of a function, which tells us about the function's rate of change or slope at any given point.

Integral calculus, on the other hand, deals with accumulation of quantities and the areas under and between curves. Here, integration plays the pivotal role, helping us understand how accumulations behave across time or other dimensions.

Techniques within calculus, such as the substitution method shown in your exercise, are vital tools in transforming complicated problems into simpler ones. These tactics reveal the interconnected nature of mathematical functions. With calculus, you're dealing with concepts that have real-world applications like physics, engineering, economics, and even biology. It teaches us how to model the world mathematically, providing insights into how complex systems evolve and interact with each other over time.