Problem 73 - Calculus: Early Transcendentals

Question

Evaluate the definite integral. \( \displaystyle \int^1_0 \frac{dx}{(1 + \sqrt{x})^4} \)

Step-by-Step Solution

Answer

The value of the integral is \( \frac{1}{2} \).

Step 1: Identify the Substitution

To evaluate the integral \( \int^1_0 \frac{dx}{(1 + \sqrt{x})^4} \), we start by using a substitution method to simplify the integrand. Let \( u = 1 + \sqrt{x} \). Then, we can express \( \sqrt{x} \) as \( u - 1 \) and consequently \( x = (u - 1)^2 \).

Step 2: Differentiate the Substitution

Differentiate \( x = (u - 1)^2 \) with respect to \( u \) to find \( dx \): \( dx = 2(u-1)du \).

Step 3: Convert Limits of Integration

Convert the original limits of integration from \( x = 0 \) to \( x = 1 \) into \( u \)-space. When \( x = 0 \), \( u = 1 + \sqrt{0} = 1 \). When \( x = 1 \), \( u = 1 + \sqrt{1} = 2 \). Thus, the integral in terms of \( u \) will be from 1 to 2.

Step 4: Substitute and Simplify the Integral

Substitute \( x = (u-1)^2 \) and \( dx = 2(u-1)du \) into the integral:\[ \int^2_1 \frac{2(u-1) \, du}{u^4} \]Simplify to:\[ 2 \int^2_1 \frac{u-1}{u^4} \, du = 2 \int^2_1 \left( \frac{1}{u^3} - \frac{1}{u^4} \right) du \].

Step 5: Integrate Each Term

Integrate each term separately:\[ 2 \left( \int^2_1 \frac{1}{u^3} \, du - \int^2_1 \frac{1}{u^4} \, du \right) \]Calculate the integrals:\( \int \frac{1}{u^3} \, du = -\frac{1}{2u^2} \)\( \int \frac{1}{u^4} \, du = -\frac{1}{3u^3} \).

Step 6: Evaluate the Definite Integral

Evaluate the definite integrals from 1 to 2:\[ 2 \left[ \left( -\frac{1}{2(2^2)} + \frac{1}{2(1^2)} \right) - \left( -\frac{1}{3(2^3)} + \frac{1}{3(1^3)} \right) \right] \].

Step 7: Calculate the Result

Compute the numerical results of the evaluated expression:\[ 2 \left[ (-\frac{1}{8} + \frac{1}{2}) - (-\frac{1}{24} + \frac{1}{3}) \right] = 2 \left[ \frac{3}{8} - \frac{1}{8} \right] = 2 \times \frac{1}{4} = \frac{1}{2} \].

Key Concepts

Integration Techniques

Integrating functions, especially in the context of definite integrals, can sometimes pose a challenge. While some integrals can be computed directly by applying basic antiderivative rules, many others require more sophisticated techniques. To simplify the integration process, mathematicians have developed various techniques. Among the well-known ones are:

• Direct Integration: Used when the antiderivative can be immediately identified.
• Substitution Method: This is used to transform the integrand into a simpler form, making it easier to evaluate. It involves substituting part of the integral with a new variable.
• Integration by Parts: Useful for products of functions, following the formula derived from the product rule of differentiation.
• Partial Fraction Decomposition: Often used when the integrand is a rational function.

In the given exercise, the Substitution Method was used to simplify the integral, transforming a challenging integrand into something more tractable for direct integration. This technique is often employed when dealing with integrals involving composite functions, such as those containing a square root or other similar transformations.

Substitution Method

The substitution method, also known as the Change of Variables, is a powerful technique utilized to solve integrals. This approach works best when the integrand contains a function nested within another function. In these cases, directly integrating can be quite complex or sometimes impossible.For the integral given in the exercise \( \int^1_0 \frac{dx}{(1 + \sqrt{x})^4} \), we identify the substitution that will simplify the expression. We let \( u = 1 + \sqrt{x} \). This choice is strategic, as it reduces the complexity of dealing with the square root inside the integrand.

Steps Involved in Substitution

• Identify Substitution: Choose a substitution that simplifies the integral. Here, we chose \( u = 1 + \sqrt{x} \).
• Express Other Variables: Convert the other parts of the integrand in terms of \( u \). Hence, \( \sqrt{x} = u - 1 \) and consequently \( x = (u - 1)^2 \).
• Differentiate: Find \( dx \) using \( u \). Differentiation gives \( dx = 2(u - 1)du \).

The substition rearranges the original integral into a new form \( \int^2_1 \frac{2(u-1) \, du}{u^4} \). This transformation reveals an integrand that can be more easily integrated.

Limits of Integration

When dealing with definite integrals, it is crucial to consider the limits of integration, as they define the interval over which you are integrating. This interval impacts the final value of a definite integral. When utilizing the substitution method, converting the limits of integration is an essential step to ensure that the integration process is carried out correctly.In the original exercise \( \int^1_0 \frac{dx}{(1 + \sqrt{x})^4} \), the limits were initially in terms of \( x \), ranging from 0 to 1. Upon applying the substitution \( u = 1 + \sqrt{x} \), these limits needed to be recalculated in terms of \( u \).Converting Limits:

• For the lower limit: When \( x = 0 \), \( u = 1 + \sqrt{0} = 1 \).
• For the upper limit: When \( x = 1 \), \( u = 1 + \sqrt{1} = 2 \).

This change of limits results in the new integral having its limits of integration from 1 to 2. Switching the limits appropriately converts the whole integration into the \( u \)-domain, allowing the calculation of the definite integral to proceed without errors.