Problem 69 - Applied Calculus

Question

Each equation follows from the integration by parts formula by replacing \(u\) by \(f(x)\) and \(v\) by a particular function. What is the function \(v ?\) \(\int f(x) d x=f(x) x-\int x f^{\prime}(x) d x\)

Step-by-Step Solution

Answer

The function \( v \) is \( x \).

Step 1: Recall Integration by Parts Formula

The integration by parts formula is given by \( \int u \, dv = uv - \int v \, du \). We need to find appropriate \( u \) and \( dv \) from the given integral.

Step 2: Assign Values to u and dv

From the given equation \( \int f(x) \, dx = f(x) x - \int x f'(x) \, dx \), we identify \( u = f(x) \) and \( dv = dx \). The corresponding choices of \( du \) and \( v \) will be determined in the next step.

Step 3: Differentiate u and Integrate dv

Differentiate \( u = f(x) \) to get \( du = f'(x) \, dx \). Integrate \( dv = dx \) to get \( v = x \).

Step 4: Verification with the Formula

Substitute \( u = f(x) \), \( du = f'(x) \, dx \), \( v = x \), and \( dv = dx \) back into the integration by parts formula: \( \int f(x) \, dx = f(x) \cdot x - \int x \, f'(x) \, dx \). This matches the given equation, confirming our choice of \( v \).

Key Concepts

Differential Calculus

Differential calculus is one of the key areas of calculus focusing on the concept of the derivative. The derivative of a function represents the rate at which the function's value changes as its input changes. This concept is fundamental when dealing with Integrals, especially in Integration by Parts. The process of differentiation involves finding the derivative, denoted by \( f'(x) \), of a function \( f(x) \). This is crucial for applications like optimization, motion analysis, and many more. In the context of Integration by Parts, differentiation helps us find \( du \), which is needed to apply the formula efficiently.

• Derivative: Symbolized as \( f'(x) \), it shows the slope of the function at a given point.
• Rate of Change: How the function's output changes as its input changes.
• Tangents and Slopes: Tangential or linear approximation of a function can be derived from the derivative.

Understanding differential calculus will enhance your skills in calculating derivatives flawlessly, making it easier to tackle more complex calculus problems.

Integration Techniques

Integration is the process of finding the integral of a function, essentially the reverse operation of differentiation. Integration techniques are essential for solving complex integrals that do not have straightforward antiderivatives. One such technique is **Integration by Parts**, which is particularly useful when dealing with products of functions. The formula \( \int u \, dv = uv - \int v \, du \) enables us to transform an integral into a simpler form.

• **Choosing \( u \) and \( dv \):** The function \( u \) is typically chosen to be easily differentiable, while \( dv \) should be easy to integrate.
• **Transforming the Integral:** Once \( u \) and \( dv \) are identified, they can be substituted back into the formula to simplify the integral.

This method is particularly effective for functions that are products, trigonometric functions, exponential functions, and logarithms.

Function Differentiation

Function differentiation is the key initial step in applying advanced integration techniques such as Integration by Parts. It begins with selecting the function to be differentiated, in this case, the function \( f(x) \) is differentiated to find \( f'(x) \). This differentiation step is critical for determining \( du \), which is a necessary component in the Integration by Parts process. The ability to quickly and accurately find derivatives is vital for successful application of integration techniques.

• Selecting \( f(x) \): A function is chosen for differentiation, ideally simple to differentiate, to facilitate the process.
• Calculating \( f'(x) \): This derivative provides the \( du \), an essential part of restructuring the integral via the Integration by Parts formula.

Being proficient in function differentiation allows one to break down complex integrals into more manageable components, aiding in accurate calculations and understanding of other calculus topics.