Problem 46 - Calculus

Question

Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{1}^{4} \frac{(\sqrt{x}-1)^{3}}{\sqrt{x}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{1}{2} \).

Step 1: Identify the Inner Function (u-substitution)

To use the substitution rule, let us first identify an inner function in the integrand. A good candidate for substitution is the expression inside the power. Let \( u = \sqrt{x} - 1 \). Thus, \( \sqrt{x} = u + 1 \).

Step 2: Find the Derivative and Differential

Differentiate both sides of the equation \( u = \sqrt{x} - 1 \) with respect to \( x \) to find \( du \). Since \( \sqrt{x} = x^{1/2} \), differentiate to get \( du/dx = \frac{1}{2}x^{-1/2} \). Therefore, \( du = \frac{1}{2\sqrt{x}} dx \), which rearranges to \( dx = 2\sqrt{x} \, du \).

Step 3: Change the Limits of Integration

Transform the limits of integration from \( x \) to \( u \) using the substitution \( u = \sqrt{x} - 1 \). When \( x = 1 \), \( u = \sqrt{1} - 1 = 0 \). When \( x = 4 \), \( u = \sqrt{4} - 1 = 1 \). Hence, the new limits are from \( u = 0 \) to \( u = 1 \).

Step 4: Substitute and Simplify the Integral

Substitute \( \sqrt{x} = u + 1 \) and \( dx = 2\sqrt{x} \, du \) into the integral. The integrand becomes:\[ \int_{0}^{1} \frac{(u)^3}{u+1} \cdot 2(u + 1) \, du = 2 \int_{0}^{1} u^3 \, du \].

Step 5: Evaluate the Integral

Now evaluate the integral \( 2 \int_{0}^{1} u^3 \, du \). First, find the antiderivative of \( u^3 \), which is \( \frac{u^4}{4} \). Thus:\[ 2 \int_{0}^{1} u^3 \, du = 2 \left[ \frac{u^4}{4} \right]_{0}^{1} = 2 \left(\frac{1^4}{4} - \frac{0^4}{4}\right) \]Evaluate this to get \( 2 \cdot \frac{1}{4} = \frac{1}{2} \).

Step 6: Final Answer

The value of the definite integral \( \int_{1}^{4} \frac{(\sqrt{x}-1)^{3}}{\sqrt{x}} \, dx \) is \( \frac{1}{2} \).

Key Concepts

u-substitution

In calculus, u-substitution is a powerful technique used to simplify integration, particularly with complex integrands. It's akin to making a variable change in order to make the integral easier to evaluate. In essence, we look inside the integral for a part that seems most complex and let it be represented by a new variable, often called "u." For instance, in the original exercise, the substitution was made by setting \( u = \sqrt{x} - 1 \). This choice simplifies the integral's complexity, transforming it to a function of \( u \). The process:

• Identify an inner function, perhaps parts of the integrand raised to a power or nested within another function.
• Set this inner function equal to \( u \).
• Differentiate to express \( dx \) in terms of \( du \), thereby simplifying substitution in the integral.

By aligning the function into a simpler form involving \( u \), we can evaluate integrals that are otherwise difficult to tackle.

limits of integration

Changing the variable in an integral also means updating the limits of integration accordingly. In definite integrals, the limits tell us over what interval to integrate. When using u-substitution, we must transform these limits based on our substitution choice. Here's how the transformation works:

• Start with the original limits in terms of \( x \), which, for the problem, are from \( x = 1 \) to \( x = 4 \).
• Convert these limits to \( u \) using the relationship established by your substitution. For \( u = \sqrt{x} - 1 \), when \( x = 1 \), \( u = 0 \); when \( x = 4 \), \( u = 1 \).
• The updated limits of integration are thus \( u = 0 \) to \( u = 1 \).

This step is crucial because integrating over incorrect limits can lead to wrong results.

antiderivative

Finding the antiderivative, or the indefinite integral of a function, is a key step in evaluating definite integrals. It's all about determining the original function given its derivative. With substitution, often the simplified integrand is more straightforward. For example, when \( u \) becomes the variable of integration, the integral might turn into something like \( \int u^3 \, du \). The antiderivative of this simpler polynomial function is \( \frac{u^4}{4} \). Steps to find an antiderivative:

• Identify the integrand after substitution; here it's \( u^3 \).
• Apply the power rule for integration: increase the exponent by one and divide by this new exponent.
• Write the antiderivative: \( \frac{u^n}{n} \), where \( n \) is the updated exponent.

This process moves us closer to evaluating the definite integral as it allows substitution of the final limits.

evaluate definite integral

With the transformation complete and the antiderivative found, it's time to evaluate the definite integral. This involves plugging in the transformed limits into the antiderivative and computing the difference. Consider the steps to complete the evaluation:

• Substitute the upper limit of integration \( u = 1 \) into the antiderivative \( \frac{u^4}{4} \), yielding \( \frac{1^4}{4} \).
• Substitute the lower limit \( u = 0 \), yielding \( \frac{0^4}{4} \).
• Compute the difference: \( \frac{1^4}{4} - \frac{0^4}{4} = \frac{1}{4} \).
• Multiply by any scalar factors outside the integral, as seen with the factor 2, to get the final answer \( \frac{1}{2} \).

This final calculation yields the precise value of the definite integral, confirming the power of the substitution method in solving complex integrals.