Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the indefinite integral. $$\int 2 u^{3 / 4} d u$$

Step-by-Step Solution

Answer

The short answer to the question 'Find the indefinite integral of \(\int 2u^{3/4} du\)' is: \(\int 2u^{3/4} du = \frac{8u^{7/4}}{7} + C\)

Step 1: Identify the function and variable

The function we are integrating is \(2u^{3/4}\), and the variable we are integrating with respect to is \(u\).

Step 2: Apply the power rule for integration

To find the indefinite integral of the function, apply the power rule for integration. Add 1 to the exponent and divide by the new exponent. So, for the function \(2u^{3/4}\), the new exponent is \(\frac{3}{4} + 1 = \frac{7}{4}\).

Step 3: Calculate the integral

Now, divide by the new exponent, \(\frac{7}{4}\), and include a constant of integration, usually denoted as \(C\). So, the indefinite integral of \(2u^{3/4}\) is: \( \int 2u^{3/4} du = \frac{2u^{7/4}}{(7/4)} + C \)

Step 4: Simplify the result

Finally, simplify our result: \( \frac{2u^{7/4}}{(7/4)} + C = \frac{8u^{7/4}}{7} + C \) So the indefinite integral of the given function is: \( \int 2u^{3/4} du = \frac{8u^{7/4}}{7} + C \)

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental technique when working with indefinite integrals. It simplifies the process of integrating functions that are in the form of a power of the variable. The power rule states that if you have a function in the form of \(u^n\), where \(n\) is any real number except \(-1\), the integral can be found by:

• Adding 1 to the exponent \(n\).
• Dividing the entire term by the new exponent \(n+1\).

This results in the formula:\[\int u^n \, du = \frac{u^{n+1}}{n+1} + C\]In our original exercise, the function \(2u^{3/4}\) fits perfectly into this rule. The exponent \(3/4\) is increased by 1, leading to \(7/4\), and then the term is divided by \(7/4\), simplifying the integration process.

Integration Techniques

Understanding various integration techniques is crucial to solving a wide range of integral problems. Besides using the power rule, you might need other techniques depending on the function's form. Some common integration techniques include:

• Substitution: Useful for integrals involving composite functions.
• Integration by parts: Often used when integrating products of functions.
• Partial fraction decomposition: Helpful for rational functions.

In our case, the power rule was the most efficient technique because the function was a simple power of \(u\). Recognizing the simplicity allowed us to quickly find the integral without additional complication.

Constant of Integration

One important aspect of indefinite integrals is the constant of integration, denoted by \(C\). When performing indefinite integration, you are essentially reversing the process of differentiation to retrieve the original function.

• The reverse process isn't unique; you can add any constant value because its derivative will be zero.
• This is why we include \(C\) in the result to account for all possible constants that could have been in the original function.

In our example, after integrating the function \(2u^{3/4}\), we add \(C\) to the final expression \(\frac{8u^{7/4}}{7} + C\). This constant makes the solution a general one, covering all potential original functions.