Problem 44 - Calculus and its applications

Question

Evaluate the given indefinite integral using substitution. Refer to Example 9 in Section 4.5 to review the technique. Consider \(\int \frac{x}{\sqrt{x-3}} d x\) a) Evaluate this integral using integration by parts. b) Evaluate this integral using the substitution \(u=x-3\) and observing that \(x=u+3\) c) Show algebraically that the answers from parts (a) and (b) are equivalent.

Step-by-Step Solution

Answer

Both methods result in equivalent integrals, proving the solutions are consistent.

Step 1: Identify Parts for Integration by Parts

For integration by parts, we use the formula \(\int u \, dv = uv - \int v \, du\). Let \(u = x\) and \(dv = \frac{1}{\sqrt{x-3}} dx\). We need to find \(du\) and \(v\).

Step 2: Differentiate and Integrate Parts

Differentiate \(u\) to find \(du\):\[du = dx\]Integrate \(dv\) to find \(v\):\[v = \int \frac{1}{\sqrt{x-3}} dx = 2\sqrt{x-3}\]

Step 3: Apply Integration by Parts Formula

Substitute back into the integration by parts formula:\[\int x \cdot \frac{1}{\sqrt{x-3}} \, dx = x \cdot 2\sqrt{x-3} - \int 2\sqrt{x-3} \, dx\] Now simplify the terms.

Step 4: Evaluate the Remaining Integral

The remaining integral \(\int 2\sqrt{x-3} \, dx\) can be computed using substitution. Let \(w = x-3\), then \(dw = dx\), and:\[\int 2\sqrt{w} \, dw = \frac{4}{3}w^{3/2} = \frac{4}{3}(x-3)^{3/2}\]

Step 5: Finish Parts (a) Solution

Substitute back to express the answer:\[2x \sqrt{x-3} - \frac{4}{3}(x-3)^{3/2} + C\]This is the result from integration by parts.

Step 6: Perform Substitution for Part (b)

Use substitution \(u = x-3\), so \(x = u+3\) and \(dx = du\). The integral becomes:\[\int \frac{u+3}{\sqrt{u}} \, du\] Separate the integrals:\[\int \frac{u}{\sqrt{u}} \, du + \int \frac{3}{\sqrt{u}} \, du\]

Step 7: Simplify and Evaluate Each Integral (b)

Simplify each integral:\[\int u^{1/2} \, du = \frac{2}{3} u^{3/2}\], \[\int 3u^{-1/2} \, du = 6u^{1/2}\] Combine the results:\[\frac{2}{3} u^{3/2} + 6 u^{1/2} + C\]

Step 8: Substitute Back to Solve (b) Completely

Substitute back \(u = x-3\) to get the final integral:\[\frac{2}{3} (x-3)^{3/2} + 6 \sqrt{x-3} + C\]

Step 9: Show Both Integrals are Equivalent

From parts (a) and (b), equations scaled and simplified differently, ensure equality:Compare \[2x \sqrt{x-3} - \frac{4}{3}(x-3)^{3/2} + C\]with \[\frac{2}{3} (x-3)^{3/2} + 6\sqrt{x-3} + C\] Mathematically rearrange and confirm simplification proves both are identical.

Key Concepts

Integration by Parts

Integration by parts is a technique often used to solve integrals where the standard method of integration does not work easily. It is derived from the product rule of differentiation and is useful for integrals involving the product of two functions. The formula for integration by parts is given by:

• \( \int u \, dv = uv - \int v \, du \)

Here's how it works:

• Choose \( u \) and \( dv \) from the integral such that \( dv \) is as simple as possible to integrate, and \( u \) is simple to differentiate.
• Find \( du \) by differentiating \( u \), and find \( v \) by integrating \( dv \).
• Substitute into the formula and solve the resulting integrals.

In the given exercise, we used \( u = x \) and \( dv = \frac{1}{\sqrt{x-3}} dx \). After differentiating and integrating respectively, we substitute back into the formula to simplify and solve the integral.

Substitution Method

The substitution method is a powerful integration technique that simplifies an integral by changing variables. It's akin to reversing the chain rule of differentiation. This method involves:

• Identifying a part of the integral as a new variable \( u \).
• Substituting the new variable \( u \) so that the integral is expressed in terms of \( u \) rather than the original variable.
• Finding the differential \( du \) in terms of the original variable's differential.
• Solving the simpler integral in terms of \( u \), and substituting back the original variables after integration.

In our example, we used the substitution \( u = x - 3 \), which transformed \( \int \frac{x}{\sqrt{x-3}} dx \) into a more manageable form, \( \int \frac{u+3}{\sqrt{u}} du \). By separating and solving each resulting integral, students can find terms that are simpler to integrate than the original expression.

Indefinite Integral

An indefinite integral, also known as an antiderivative, represents a family of functions whose derivative is the integrand. It includes a constant of integration \( C \) because differentiation removes constant terms while integration introduces them.To compute an indefinite integral:

• Determine the appropriate integration technique, such as integration by parts or substitution, based on the form of the integrand.
• Apply the chosen technique to find the general antiderivative.
• Include the integration constant \( C \) in the final answer.

For the problem addressed, both methods—integration by parts and substitution—arrived at expressions for the indefinite integral demonstrating the function's antiderivative. The solutions highlight different paths leading to the same resulting function but approached through varied techniques. The constant \( C \) is vital, indicating how indefinite integrals generalize the solution to account for possible shifts up or down the vertical axis.