Question

Find each indefinite integral. Check some by calculator. $$\int d y$$

Short answer

\( y + C \)

Step-by-step solution

Understand the Indefinite Integral of dy

The given indefinite integral \(\int d y\) asks us to find the function whose derivative with respect to y is 1. Since the derivative of a variable with respect to itself is 1, we can infer that the integral of \(d y\) represents the antiderivative of 1 with respect to y.

Find the Antiderivative

The antiderivative of a constant is the constant times the variable of integration plus a constant of integration (since the derivative of any constant is zero). Here, the antiderivative of 1 with respect to y is simply y, and we add a constant term \(C\) to account for all potential antiderivatives.

Write the Final Result

Thus, the indefinite integral of \(d y\) is \[ \int d y = y + C \.\]

Key concepts

Antiderivative

When solving for an antiderivative, we seek a function akin to reversing the process of differentiation. Think of it as detective work, where instead of identifying the result of a change (derivative), we're interested in discovering the origin of that change. In the exercise, the quest is to unearth a function whose derivative equals 1. Since taking the derivative of y with respect to itself results in 1, we've found that the antiderivative in this scenario is simply y. Integration is the broader branch of mathematics that encompasses finding antiderivatives, and as we delve into exercises like the one given, we're practicing the foundational skills of integral calculus.

An antiderivative is not unique; it's a family of functions due to the fact that differentiation wipes out constant values. That's where the importance of adding a constant term, known as the constant of integration, comes into play, ensuring we consider all possible original functions.

Integral Calculus

Within the realm of integral calculus, we visit two main theatres: indefinite and definite integrals. The challenge we faced in this exercise is an example of an indefinite integral, where we don't have specific limits of integration and are instead presented with a general form. Integral calculus allows us to find the accumulated totals, such as areas under curves, and to solve problems of total change by reinstating what was lost during differentiation, the constants.

Integral calculus is not merely a set of mathematical techniques; it is a language through which we interpret growth, areas, and much more in various fields. The process of integrating fosters a comprehensive understanding of how functions accumulate their values – a critical concept for students in not just math, but also physics, engineering, economics, and beyond.

Constant of Integration

The constant of integration, denoted typically by the symbol C, is an essential aspect of the indefinite integral. When we differentiate a function, we lose any constant terms since their rate of change is zero. Therefore, when we reverse this process through integration, we need to acknowledge the existence of an unknown constant.

Given the task of integrating dy, we've concluded that y is an antiderivative. However, to account for all possible functions that could differentiate to give us 1, we must include the constant C. This C represents any and all constants that were wiped clean during differentiation, making the family of antiderivatives complete. It is crucial to include this constant to cover the full spectrum of solutions and to achieve accuracy in mathematical expressions that describe real-world phenomena.