Problem 11 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{x^{2}-4}{x} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the given function is \( \frac{1}{2}x^2 - 4 ln|x| + C \).

Step 1: Simplify the Integral

We begin by simplifying the integral by dividing each term in the numerator by the denominator separately. This simplification will result in the following: \[ \int \frac{x^2}{x} dx - \int \frac{4}{x} dx \].

Step 2: Simplify Further

We can simplify the divided terms further to get \[ \int x dx - \int 4 \frac{1}{x} dx \] This gives us two simpler integrals to solve.

Step 3: Find the Antiderivative

Next we find the integrals of the two simpler functions. The antiderivative of \( x \) with respect to \( x \) is \( \frac{1}{2}x^2 \) and the antiderivative of \( \frac{1}{x} \) with respect to \( x \) is \( ln|x| \). So, we calculate the integrals: \[ \int x dx = \frac{1}{2}x^2 + C_1 \] \[ \int 4 \frac{1}{x} dx = 4 \int \frac{1}{x} dx = 4 ln|x| + C_2 \]

Step 4: Combine Results

The original function therefore resolves into these two functions added together: \[ \frac{1}{2}x^2 + C_1 - 4 ln|x| - C_2 \] Because \( C_1 \) and \( C_2 \) are both constants, we can combine them into a single constant \( C \). This leads us to the final solution.

Key Concepts

Antiderivative

When finding the indefinite integral of a function, we actually search for its antiderivative. An antiderivative of a function is another function whose derivative gives back the original function. It's like working backwards in calculus, moving from a derivative back to its original form.The connection between an indefinite integral and finding an antiderivative is profound yet straightforward. The integral sign \( \int \) accompanies our function, and when we determine its indefinite integral, we actually find the collection of all possible antiderivatives.For example, in the given exercise, breaking down the function \( \int x \, dx \) results in an antiderivative of \( \frac{1}{2}x^2 \). Similarly, the antiderivative of \( \int \frac{1}{x} \, dx \) is \( \ln|x| \). These expressions reveal how the process of integration "undoes" differentiation. This is why, when solving indefinite integrals, a key task is to determine the function that, when derived, returns the integrand.

Simplification of Fractions

Simplifying fractions is a crucial skill, particularly in calculus, to make integrals more manageable. The initial step in solving indefinite integrals often involves breaking down complex expressions into simpler parts.Think of this process as reorganizing or rewriting the integral to make it easier to handle. For example, in the exercise \( \int \frac{x^2 - 4}{x} \, dx \), simplifying \( \frac{x^2 - 4}{x} \) requires dividing each term of the numerator by the denominator \( x \). This results in two separate terms: \( \frac{x^2}{x} = x \) and \( \frac{-4}{x} = -4\frac{1}{x} \).

• This separation transforms the integral into a form that's simpler to integrate, namely \( \int x \, dx - \int 4 \frac{1}{x} \, dx \).
• Such decomposition not only makes handling the integral easier but also enhances our understanding of the function's behavior.

Constant of Integration

Whenever we find an indefinite integral, a constant of integration, denoted usually by "\( + C \)", must be included. This constant represents the family of all potential antiderivatives for a given function.Integrals provide antiderivatives up to a constant because differentiation eliminates constants. Therefore, to account for all possible shifts in the antiderivative graph, we attach \( C \).While managing constants in definite integrals often involves cancellation, indefinite integrals maintain this constant. In our problem, after integrating, results like \( \frac{1}{2}x^2 + C_1 \) and \( 4 \ln|x| + C_2 \) carry constants \( C_1 \) and \( C_2 \).

• Combining these into a single constant \( C \) is plausible because the "shift" they represent doesn't affect the indefinite integral's correctness.
• The solution therefore simply integrates these as one unified constant for clarity and simplicity.

Including the constant of integration ensures all solutions within the set of antiderivatives are represented. It is a small yet critical detail that should always accompany any established antiderivative.