Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Use any basic integration formula or formulas to find the indefinite integral. State which integration formula(s) you used to find the integral. $$ \int \frac{1}{\sqrt{x+1}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(\frac{1}{\sqrt{x+1}}\) with respect to \(x\) is \(2 \sqrt{x+1} + C\).

Step 1: Identify the Function Type

Looking at the integrand, \(\frac{1}{\sqrt{x+1}}\), it can be recognized as a function in the form of \(\frac{1}{\sqrt{a(x)}}\), where \(a(x) = x+1\). It is also important to note that the exponent of \(a(x)\) is \(-\frac{1}{2}\). So the function is similar to \(\int a(x)^n dx \) where \(n=-\frac{1}{2}\).

Step 2: Apply the Basic Integration Formula

The basic integration formula needed in this situation is \(\int a(x)^n dx = \frac{a(x)^{n+1}}{n+1} + C \), where \(C\) is the constant of integration, \(a(x)\) is a function of \(x\), and \(n\) is the power of \(a(x)\). Now we substitute \(a(x)=x+1\) and \(n=-\frac{1}{2}\) into the formula, and integrate.

Step 3: Perform The Integration

Using the formula, the integration is as follows. \(\int \frac{1}{\sqrt{x+1}} dx = \frac{(x+1)^{-1/2+1}}{-1/2+1} + C = \frac{(x+1)^{1/2}}{1/2} + C = 2 \sqrt{x+1} + C\).

Key Concepts

Integration Formulas

The cornerstone of integral calculus lies in the array of integration formulas that help us solve a wide variety of integrals. Understanding these formulas can be likened to having a toolbox where each tool serves a specific purpose. For instance, consider the exercise where we are required to find the indefinite integral of a function written as \(\int \frac{1}{\sqrt{x+1}} dx\). The choice of formula here is instrumental. Integrals of the type \(\int a(x)^n dx\) where \( a(x) \) is a function of \( x \) and \( n \) is a numerical exponent, can be tackled with the formula \(\frac{a(x)^{n+1}}{n+1} + C \), as long as \( n \) is not -1. In this example, the formula simplifies the process significantly by providing a direct method to integrate powers of \( x \).
Integration strategies often start with identifying whether the integral matches one of these fundamental formulas. Once identified, we can proceed with confidence, knowing that we have the correct tool for the job.

Integral Calculation

After identifying the correct integration formula, we proceed with integral calculation. This involves replacing the variable parts of the formula with the specific components of the function we are integrating. In our case, \( a(x) = x + 1 \) and \( n = -\frac{1}{2} \). Substituting these values into the formula \(\int a(x)^n dx = \frac{a(x)^{n+1}}{n+1} + C\), we perform the algebraic operations to solve for the integral.
Integral calculation requires careful manipulation of algebra, especially with regard to exponents and coefficients. Performing these steps, we find that \(\int \frac{1}{\sqrt{x+1}} dx = 2\sqrt{x+1} + C\). This step-by-step approach showcases the importance of algebraic skills when working with integrals, ensuring each calculation is done with precision and care.

Constant of Integration

When we find the indefinite integral of a function, the result includes a constant of integration, denoted by \( C \). This constant is inherently connected to the concept of an indefinite integral, which represents a family of functions rather than a single unique solution. This is because differentiation of a constant results in zero, meaning any constant could have been present in the original function prior to its differentiation.
The inclusion of \( C \) in our final answer \(2\sqrt{x+1} + C\) accounts for all possible antiderivatives of the given function. This is a crucial part of the process and is often a point where mistakes can be made if overlooked. Understanding the role of the constant of integration ensures that students can interpret the results of an indefinite integral correctly and fully appreciate the subtleties of integral calculus.