Problem 3

Question

Find each integral. $$ \int 2 d x $$

Step-by-Step Solution

Verified

Answer

The integral is $ 2x + C $.

1Step 1: Identify the Integral Type

The given integral $ \int 2 \, dx $ is a basic indefinite integral, where we need to find the antiderivative of a constant function.

2Step 2: Integral of a Constant Function

To solve $ \int 2 \, dx $, recognize that the antiderivative of a constant $ c $ with respect to $ x $ is $ cx $. So, the antiderivative of 2 is $ 2x $.

3Step 3: Add the Constant of Integration

Since this is an indefinite integral, we add a constant $ C $ to represent the family of all antiderivatives. Therefore, the integral becomes $ 2x + C $.

Key Concepts

Indefinite IntegralAntiderivativeConstant of Integration

Indefinite Integral

An indefinite integral, denoted as $ \int f(x) \, dx $, is a fundamental concept in calculus that represents the collection of all antiderivatives of a function. Unlike the definite integral, which results in a number, an indefinite integral results in a function. This means that instead of calculating the area under a curve for a specific interval, we are looking to find a family of functions whose derivative is the original function given in the integral.When dealing with an indefinite integral, we don't have specific limits of integration. We focus on reversing the differentiation process. Essentially, it's about "undoing" the derivative.

Indefinite integrals include a constant of integration, $ C $, which accounts for all possible vertical shifts of the function.
For example, the indefinite integral of $ 2 $ with respect to $ x $ is $ 2x + C $.

This represents all possible functions whose derivatives would yield $ 2 $. The process involves basic antiderivative rules which can be applied to numerous functions.

Antiderivative

The antiderivative is essentially the opposite of differentiation. When we differentiate a function, we find its rate of change. Conversely, when we find an antiderivative, we seek a function whose derivative matches the given function.An antiderivative is sometimes called a primitive function. It's like working backward in a puzzle to discover which function could have produced the given rate of change.

If $ F'(x) = f(x) $, then $ F(x) $ is an antiderivative of $ f(x) $.
For a constant function, such as $ 2 $, the antiderivative would be $ 2x $ because the derivative of $ 2x $ is $ 2 $.

There can be infinitely many antiderivatives differing by a constant, which can all be captured by adding a constant of integration, $ C $. This leads us to the next critical concept in indefinite integrals.

Constant of Integration

In the realm of indefinite integrals, every antiderivative includes a constant of integration, denoted by $ C $. This essential component embodies the infinite number of vertical shifts of the antiderivative along the y-axis, retaining the same slope.The reason we add $ C $ is because when a function is differentiated, any constant is reduced to zero. Thus, when you integrate and find all possible antiderivatives, there's no way to know which constant was in the original function.

For example, considering the integral $ \int 2 \, dx = 2x + C $, the $ C $ accounts for any missing constant from when the derivative was taken.
This ensures the general solution of the integral rather than a single particular solution.

Remembering to include $ C $ is crucial for correctly solving any indefinite integral problem.

Problem 3

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