Question

Find each indefinite integral. Check some by calculator. $$\int 7 \, d y$$

Short answer

\(\int 7 \, dy = 7y + C\)

Step-by-step solution

Understand the Indefinite Integral of a Constant

The indefinite integral of a constant, say 'c', with respect to a variable, say 'y', is simply the product of the constant and the variable, plus an arbitrary constant 'C' because the antiderivative is not unique. In this case, the constant is 7. We express this as \(\frac{d}{dy}(7y + C) = 7\). The integral you are trying to solve is \(\int 7 \, dy\).

Integrate the Constant

The integral of a constant 'a' with respect to a variable 'x' is 'ax', plus an arbitrary constant 'C'. Apply this to \(7\) with respect to \(dy\): \(\int 7 \, dy = 7y + C\), where 'C' is the constant of integration.

Checking the Result with a Calculator

To check your work, you can differentiate your result with respect to 'y'. You should end up with the original function. Differentiate \(7y + C\) with respect to 'y' to confirm that you get \(7\). Some advanced calculators or software can directly compute the indefinite integral. Use one such tool to validate the solution: it should give you the same result, possibly with a different notation for the constant.

Key concepts

Antiderivative

Understanding the concept of an antiderivative is vital for grasping the fundamentals of integral calculus. An antiderivative of a function is another function that, when differentiated, yields the original function. Think of it like solving a mystery where you’re given the outcome and you need to find out what caused it. The process of finding an antiderivative is known as integration, and it is as crucial to calculus as operations like addition and subtraction are to basic arithmetic.

For example, if you're working with the function given by the derivative, say, \(f'(y) = 7\), the goal is to figure out a function \(f(y)\) whose derivative is 7. This function \(f(y)\) is an antiderivative of \(f'(y)\). There are actually many functions that could be the antiderivative of \(f'(y)\), since adding any constant to \(f(y)\) will not change its derivative. Such functions are expressed in the form \(f(y) = 7y + C\), where \(C\) represents the constant of integration, indicating that many antiderivatives are possible.

Constant of Integration

When we integrate a function, we are essentially adding up an infinite number of infinitesimally small pieces to find the total accumulation. But because differentiation wipes out any constant term (since the derivative of a constant is zero), when we reverse the operation by integrating, we must add a 'constant of integration' to account for any constant that may have been lost in the process of differentiation.

This constant, typically denoted as \(C\), serves as a placeholder for all the infinite constant values that could be added to the antiderivative without affecting its derivative. It's like adding a 'C' to every indefinite integral is like saying, 'There's a piece of the puzzle that might be missing, but it doesn't change the overall picture created by the function's rate of change.' Remember, this constant only appears when you're working with indefinite integrals. For definite integrals, which have clear boundaries, the constant cancels out because you are finding the net area between the function and the x-axis.

Integrating a Constant

Integrating a constant might seem like one of the simpler operations in calculus, yet it's a fundamental task that offers insight into the principles of integration. When you integrate a constant with respect to a variable, you're essentially determining the antiderivative of that constant. It's akin to saying, 'If I travel at a constant speed, how far will I have gone after a certain amount of time?'. The integral of a constant \(a\) with respect to a variable \(x\) is a straight line represented by the equation \(ax + C\), where \(C\) is the constant of integration we've discussed earlier.

The operation is straightforward: \(\int a \, dx = ax + C\). When dealing with \(\int 7 \, dy\), you think, 'What function, when differentiated with respect to \(y\), gives me 7?' The answer is a linear function \(7y\), with the constant of integration added: \(7y + C\). The simplicity of this concept allows for a solid foundation as the integrals become more complex, involving variables and more intricate functions.