Problem 52 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int\left(3 x^{1 / 3}+\frac{1}{3 x^{1 / 3}}\right) d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{9}{4}x^{4/3} + \frac{1}{2}x^{2/3} + C \).

Step 1: Separate the Integral

We can split the given integral into two separate integrals as follows: \[ \int \left( 3x^{1/3} + \frac{1}{3x^{1/3}} \right) dx = \int 3x^{1/3} \, dx + \int \frac{1}{3x^{1/3}} \, dx \] This allows us to integrate each term individually.

Step 2: Simplify Each Integral

Rewrite each part of the integral for easier integration: \[ \int 3x^{1/3} \, dx = 3 \int x^{1/3} \, dx \] and \[ \int \frac{1}{3x^{1/3}} \, dx = \frac{1}{3} \int x^{-1/3} \, dx \]. By factoring out constants, each integral is simplified.

Step 3: Integrate Each Term

Apply the power rule for integration, which is \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]:For the first integral:\[ \int x^{1/3} \, dx = \frac{x^{1/3 + 1}}{1/3 + 1} = \frac{x^{4/3}}{4/3} = \frac{3}{4}x^{4/3} \] So, \[ 3 \int x^{1/3} \, dx = 3 \times \frac{3}{4}x^{4/3} = \frac{9}{4}x^{4/3} \]For the second integral:\[ \int x^{-1/3} \, dx = \frac{x^{-1/3 + 1}}{-1/3 + 1} = \frac{x^{2/3}}{2/3} = \frac{3}{2}x^{2/3} \] Thus,\[ \frac{1}{3} \int x^{-1/3} \, dx = \frac{1}{3} \times \frac{3}{2}x^{2/3} = \frac{1}{2}x^{2/3} \].

Step 4: Combine the Results

Combine the results of the individual integrals: \[ \int 3x^{1/3} \, dx + \int \frac{1}{3x^{1/3}} \, dx = \frac{9}{4}x^{4/3} + \frac{1}{2}x^{2/3} + C \] where \(C\) is the constant of integration. This is the final solution.

Key Concepts

Power Rule for Integration

The power rule for integration is one of the fundamental techniques used in calculus for finding indefinite integrals. It is analogous to the power rule for derivatives. The power rule states: \[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where \(n\) is a real number different from -1, and \(C\) is the constant of integration. It's a straightforward method that allows you to handle integrals involving monomials or terms with variables raised to a power.In the example exercise, we first encounter terms like \(x^{1/3}\) and \(x^{-1/3}\). Using the power rule, they are integrated separately after rewriting constants outside the integral to simplify computation. Remembering the formula is essential for tackling any basic power of \(x\) in integration tasks.

Integration Techniques

There's a suite of techniques that can be used to solve integration problems, helping to simplify complex integrals into more manageable parts. One primary strategy is to split the integral into separate terms, which was effectively employed in our exercise.Let's break it down further:

• First, separate the terms inside the integral: \( \int \left( 3x^{1/3} + \frac{1}{3x^{1/3}} \right) dx = \int 3x^{1/3} \ dx + \int \frac{1}{3x^{1/3}} \ dx \). This allows us to deal with each term independently.
• Next, simplify by factoring out constants such as 3 and \(\frac{1}{3}\), ensuring the integrals are simpler to evaluate individually.

These techniques are instrumental in managing more elaborate integrals by breaking them down into simpler parts that are easier to evaluate.

Calculus Problem Solving

Solving calculus problems, especially indefinite integrals, involves a careful step-by-step approach. Let's review the process applied to the example:

• **Identify and separate**: Spot each term and separate the integral if necessary. This organizes the problem into smaller, clear sections.
• **Manipulate and simplify**: Rearrange terms, factor out constants, and adjust exponents to utilize known integration rules, making the problem more approachable.
• **Apply integration rules**: Use techniques like the power rule for each individual component.
• **Combine results**: After integrating each term, combine them back together, including the constant of integration \(C\), which accounts for any potential offset in solutions.

By methodically following these steps, students see the power of systematic problem-solving in calculus, allowing them to tackle a wide array of problems with confidence.