Problem 53 - Calculus Single Variable

Question

Calculate the integrals. $$ \int x \cdot(2 x+3)^{1 / 2} d x $$

Step-by-Step Solution

Answer

\[ \int x \cdot (2x+3)^{1/2} \, dx = \frac{1}{5} (2x+3)^{5/2} - \frac{1}{2} (2x+3)^{3/2} + C \]

Step 1: Identify the Integral Type

The integral \( \int x \cdot (2x + 3)^{1/2} \, dx \) involves a product of functions. We have an algebraic expression \( x \) and a radical expression \( (2x + 3)^{1/2} \), suggesting substitution might be useful.

Step 2: Choose a Substitution

Choose a substitution that simplifies the expression under the radical. Let \( u = 2x + 3 \). Then, compute the differential: \( du = 2 \, dx \). This implies that \( dx = \frac{1}{2} du \). Also, express \( x \) in terms of \( u \): \( x = \frac{u - 3}{2} \).

Step 3: Rewrite the Integral

Substitute all occurrences of \( x \) and \( dx \) in the integral: \[ \int \left(\frac{u - 3}{2}\right) \cdot u^{1/2} \cdot \frac{1}{2} \, du = \frac{1}{4} \int (u - 3) u^{1/2} \, du\]

Step 4: Expand the Product

Expand the product \( (u - 3) u^{1/2} = u^{3/2} - 3u^{1/2} \).The integral becomes: \[ \frac{1}{4} \int (u^{3/2} - 3u^{1/2}) \, du\] which can be split as \[ \frac{1}{4} \left(\int u^{3/2} \, du - 3 \int u^{1/2} \, du \right)\]

Step 5: Integrate Each Term

Use the power rule for integration \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \):- For \( u^{3/2} \): \( \int u^{3/2} \, du = \frac{u^{5/2}}{5/2} = \frac{2}{5} u^{5/2} \).- For \( u^{1/2} \): \( \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3} u^{3/2} \).The integrals become: \[ \frac{1}{4} \left( \frac{2}{5} u^{5/2} - 3 \cdot \frac{2}{3} u^{3/2} \right)\]

Step 6: Simplify and Substitute Back

Simplify the coefficients and substitute back \( u = 2x + 3 \):\[ = \frac{1}{4} \left( \frac{2}{5} (2x + 3)^{5/2} - 2 (2x + 3)^{3/2} \right)\]\[ = \frac{1}{5} (2x + 3)^{5/2} - \frac{1}{2} (2x + 3)^{3/2} + C\]Finally, we obtain the integral result.

Key Concepts

Substitution Method in Integration

The substitution method is a powerful tool in integration. It is particularly useful when dealing with integrals involving complex expressions, such as products of functions. For instance, in the integral \( \int x \cdot (2x+3)^{1/2} \, dx \), the substitution helps to simplify the problem by transforming it into a more manageable form. Here's how it works:

• Identify a part of the integrand to substitute. Usually, this is the most complicated part, like a radical or an exponential function.
• Define a new variable for this part, like \( u = 2x + 3 \). This transforms the complex expression into a simpler one.
• Find the differential \( du \) and express \( dx \) in terms of \( du \) to replace the differentials in the integral.
• Rewrite the integral in terms of \( u \), which simplifies the integration process.

This method not only simplifies the computation but also often reveals the underlying structure of the integral that was not apparent at first glance.

Power Rule for Integration

The power rule for integration is one of the simplest yet most important tools in calculus. It applies to integrals of the form \( \int u^n \, du \), where \( n eq -1 \). In the step-by-step solution of integrating \( \int x \cdot (2x+3)^{1/2} \, dx \), this rule is employed after substitution. The power rule states:

• For any real number \( n \), \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \),
• where \( C \) is the integration constant.

In our case, after substituting \( u = 2x + 3 \), we have terms like \( u^{3/2} \) and \( u^{1/2} \). The power rule can be directly applied:

• \( \int u^{3/2} \, du = \frac{u^{5/2}}{5/2} = \frac{2}{5}u^{5/2} \)
• \( \int u^{1/2} \, du = \frac{u^{3/2}}{3/2} = \frac{2}{3}u^{3/2} \)

It's a straightforward process that highlights the importance of the power rule in calculus by turning complex polynomials into easily integrable expressions.

Product of Functions in Integrals

Integrating products of functions, such as \( x \cdot (2x+3)^{1/2} \), can be particularly challenging. When you encounter a product in an integral, it requires strategic thinking to simplify and solve. One effective strategy is to manipulate the integrand using techniques like substitution, making the product easier to integrate. Here's a breakdown:

• You first identify if one of the functions within the product can be simplified through substitution. Here, we used \( u = 2x + 3 \). This substitution encapsulated the complexity of the radical and hid it in a single variable.
• After substitution and expressing the original integral in terms of \( u \), often, the integral can be split and expanded, simplifying the product into a sum of terms that can be integrated individually.
• This approach typically results in each term being easier to integrate, as seen when separating into \( u^{3/2} \) and \( u^{1/2} \).

By skillfully breaking down the product into simpler parts, along with using the substitution and power rule, this method effectively simplifies complex integrals involving products of functions.