Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to find the indefinite integral. Verify the result by differentiating. $$ \int \frac{x}{\sqrt{3 x+2}} d x $$

Step-by-Step Solution

Answer

The indefinite integral of the function \( \frac{x}{\sqrt{3x+2}} \) is \( \frac{2}{5}(3x+2)^{5/2} - 4(3x+2)^{1/2} + C \) and it is verified by differentiation.

Step 1: Perform the Integration

In order to find the indefinite integral of the given function, \( \frac{x}{\sqrt{3x+2}} \), you need to use symbolic integration. Applying the principles of calculus, mainly integration by substitution, you get:For the substitution, let \( u = 3x + 2 \). Then, \( du = 3dx \) and \( dx = \frac{1}{3}du \). Now, substitute these values into the integral:\[ \int \frac{x}{\sqrt{u}} * \frac{1}{3} du \]This simplifies to \[ \frac{1}{3} \int \frac{u-2}{\sqrt{u}} du \]which can be distributed as \[ \frac{1}{3} \int \left( u^{1/2} - 2u^{-1/2} \right) du \]The integral can be evaluated to \[ \frac{2}{5}u^{5/2} - 4u^{1/2} + C \]Finally, substitute \( u \) back with \( 3x + 2 \):\[ \frac{2}{5}(3x+2)^{5/2} - 4(3x+2)^{1/2} + C \]

Step 2: Verifying the result by differentiation

To verify the correctness of the integral, it's needed to differentiate the function and compare it to the original function. The derivative is as follows:\[ \frac{d}{dx} \left( \frac{2}{5}(3x+2)^{5/2} - 4(3x+2)^{1/2} \right) \]Which simplifies to\[ \frac{x}{\sqrt{3x+2}} \]Seeing that the derivative matches the original equation, it can be concluded that the answer is correct.

Key Concepts

Symbolic Integration

Symbolic integration is a method used in calculus to determine the indefinite integral of a function. When tackling an integral symbolically, it means we are seeking an expression in terms of simple functions without computing any numerical value. In the exercise, we are asked to symbolically integrate \( \int \frac{x}{\sqrt{3x+2}} \, dx \). This involves finding a function whose derivative is the given integrand. Unlike numerical integration, which involves approximating the area under a curve, symbolic integration provides an exact expression.

• Helps us transform complex expressions into simpler forms.
• Can be performed using various methods, such as substitution or integration by parts.
• Involves algebraic manipulation to simplify the result, as seen with the expression \( \int \frac{x}{\sqrt{u}} \, du \).

Using a symbolic integration utility, we apply calculus concepts to perform the integration and find the antiderivative of the function, which includes constants represented by \( C \), representing the integral's indefinite nature.

Integration by Substitution

Integration by substitution is a powerful technique in calculus used to simplify and solve integrals. It is especially useful when dealing with composite functions, where direct integration is challenging. The core idea is to "substitute" part of the integrand with a new variable, making the integral easier to evaluate. In our problem, we substitute \( u = 3x + 2 \), turning the original integral into one in the variable \( u \), which simplifies our integration process.

• We selected \( u = 3x + 2 \) because the derivative \( du = 3dx \) matches a part of the integrand.
• By rewriting the integral in terms of \( u \), we simplified it to \( \int (u^{1/2} - 2u^{-1/2}) \, du \).
• This allowed us to integrate each term separately, making it straightforward to apply standard integration rules.

After performing the integration with respect to \( u \), we substitute back \( u = 3x + 2 \) to express the antiderivative in terms of \( x \). This step reconnects our result to the original variable, completing the integration by substitution process.

Differentiation Verification

Differentiation verification is a method to check the correctness of an obtained indefinite integral. By differentiating the result and comparing it to the original integrand, we confirm that our integration was accurate. In this exercise, after obtaining the antiderivative \( \frac{2}{5}(3x+2)^{5/2} - 4(3x+2)^{1/2} + C \), we perform differentiation to see if it matches the original function \( \frac{x}{\sqrt{3x+2}} \).

• Differentiation is the reverse process of integration, hence it's used for verification.
• By applying derivative rules, we calculate the derivative of the antiderivative.
• Matching the derivative with the original integrand confirms the correctness of the integration.

In our solution, the differentiation yields \( \frac{x}{\sqrt{3x+2}} \), confirming that the symbolic integration was performed correctly. This verification step ensures the accuracy and reliability of our integral solution, providing confidence in our obtained result.