Problem 62 - Thomas Calculus in SI Units

Question

Verify the formulas by differentiation. $$\int \frac{1}{(x+1)^{2}} d x=\frac{x}{x+1}+C$$

Step-by-Step Solution

Answer

Differentiation confirms the formula is correct.

Step 1: Understand the Problem

We need to verify if the given integral formula, \( \int \frac{1}{(x+1)^{2}} \, dx = \frac{x}{x+1}+C \), is true by differentiating the right-hand side and checking if it equals the integrand, \( \frac{1}{(x+1)^{2}} \).

Step 2: Differentiate the Right-Hand Side

The given formula is \( \int \frac{1}{(x+1)^{2}} \, dx = \frac{x}{x+1}+C \).Differentiate \( \frac{x}{x+1} \) with respect to \( x \). We apply the quotient rule:\( \frac{d}{dx} \left( \frac{x}{x+1} \right) = \frac{(x+1)(1) - x(1)}{(x+1)^2} \).Simplify the expression:\( \frac{1}{(x+1)^2} \).

Step 3: Compare the Results

The derivative of the right-hand side, \( \frac{x}{x+1} \), is \( \frac{1}{(x+1)^2} \), which matches the integrand.This confirms the integration result. Thus, the given formula is verified as correct.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the rate at which a function changes at any given point. This rate is often referred to as the derivative of the function. In simple terms, when you differentiate a function, you are looking for its slope. Derivatives help us understand how a function behaves. - For example, they can tell us how fast something is moving in a physics problem, or how quickly temperatures are changing in a weather model. - The derivative gives us a new function, which can be used for further analysis, like finding maximum or minimum points of the original function. In this particular exercise, differentiation was used to verify that the integration result was correct by checking if the derivative of the integrated function matched the original function within the integral.

Quotient Rule

When differentiating functions, especially those that are expressed as fractions, the quotient rule is a valuable tool. The quotient rule is applied when calculating the derivative of a division of two functions.The formula for the quotient rule is:\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \]where \(u\) and \(v\) are functions of \(x\), and \(u'\) and \(v'\) are their respective derivatives.This rule is particularly helpful in this exercise. By using the quotient rule, we differentiate the function \( \frac{x}{x+1} \), which helped confirm the integration result. This application involves simplicity, because the function in the numerator is a straightforward \(x\), making the differentiation more manageable. Remember, the key with the quotient rule is ensuring accuracy in applying the rule: multiply, subtract, and then divide as per the formula.

Definite and Indefinite Integrals

Integrals in calculus can either be definite or indefinite. Indefinite integrals represent a family of functions and include a constant \(C\) because differentiation of constants is zero and cannot be recovered by integration alone. The format looks like this: \[ \int f(x) \, dx = F(x) + C \]where \(F(x)\) is the antiderivative of \(f(x)\).Definite integrals, on the other hand, provide a specific numerical value representing the area under the curve of a function between two points, \(a\) and \(b\). It’s symbolized by:\[ \int_a^b f(x) \, dx \]This contrasts with indefinite integrals, which do not have specified limits of integration.In the original exercise, we deal with an indefinite integral, \( \int \frac{1}{(x+1)^2} \, dx \), which results in a function plus a constant \(C\). Verification through differentiation checks that the calculated antiderivative works correctly, highlighting the importance of understanding both definite and indefinite integrals for solving problems in calculus.