Problem 48 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Integral Type

We are given the integral \( \int (1+x^3) \sqrt{x} \, dx \). This integral involves a polynomial \((1+x^3)\) and \(\sqrt{x}\). We can use the distributive property and focus on simplifying the function inside the integral first.

Step 2: Distribute the Terms

Distribute \(\sqrt{x} = x^{1/2}\) across each term in \((1 + x^3)\). Rewrite the expression as: \((1 + x^3)x^{1/2} = x^{1/2} + x^{3+1/2} = x^{1/2} + x^{7/2}\).

Step 3: Break Down the Integral

Divide the integral into simpler parts: \( \int (x^{1/2} + x^{7/2}) \, dx = \int x^{1/2} \, dx + \int x^{7/2} \, dx \). We will solve each part separately.

Step 4: Solve Each Integral

For \(\int x^{1/2} \, dx\), use the power rule for integration. The power rule states \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\), where \(n eq -1\).- For \(x^{1/2}\): \( n = \frac{1}{2} \) \[ \int x^{1/2} \, dx = \frac{x^{1/2+1}}{1/2+1} = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2} \]- For \(x^{7/2}\): \( n = \frac{7}{2} \) \[ \int x^{7/2} \, dx = \frac{x^{7/2+1}}{7/2+1} = \frac{x^{9/2}}{9/2} = \frac{2}{9}x^{9/2} \]

Step 5: Combine Results and Include Constant of Integration

Combine the results from Step 4, remembering to add the constant of integration \(C\), which is a rule for indefinite integrals.\[ \int x^{1/2} + x^{7/2} \, dx = \frac{2}{3}x^{3/2} + \frac{2}{9}x^{9/2} + C \]

Step 6: Finalize the Solution

Combine the terms to present the solution for the initial integral expression:\[ \int (1+x^3)\sqrt{x} \, dx = \frac{2}{3}x^{3/2} + \frac{2}{9}x^{9/2} + C \]

Key Concepts

Power Rule for Integration

The power rule is a foundational tool in calculus that simplifies the process of integration, particularly when dealing with polynomials. If you understand how to apply the power rule, you will find it much easier to integrate simple expressions involving powers of x. Here's how the power rule for integration works: for any term of the form \(x^n\) where \(neq -1\), you can integrate it by increasing the exponent by 1 and then dividing by the new exponent. So, mathematically, it is expressed as:

• \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)

This rule is particularly useful when you encounter terms like \(x^{1/2}\) or \(x^{7/2}\). Simply replace \(n\) with the number that's the power of \(x\), adjust the exponent, divide by this new exponent, and add the constant \(C\) to account for indefinite integration.Using the power rule correctly allows for tackling more complex expressions by breaking them down into simpler pieces, each representing a straightforward polynomial integration problem.

Integration Techniques

Integration isn't just about applying a single formula; sometimes, you require a strategy to simplify the expressions before integrating. One effective method is simplifying the expression itself. In our example of \(\int (1+x^3) \sqrt{x} \, dx\), it involves

• Distributing: First, recognize that some terms, like polynomials, can be simplified by distributing other terms across them. This helps you rewrite the integral in a simpler form.
• Substitution: Another method, not used here but worth noting, is substitution, where you replace complex parts of an integral with a single variable to ease calculation.

In our integral, we rewrote \(\sqrt{x}\) as \(x^{1/2}\) and distributed it across \((1+x^3)\), simplifying our work down to integrating \(x^{1/2}\) and \(x^{7/2}\) separately. This not only makes the process simpler but also clearer, ensuring each part of the expression is straightforwardly integrable using the power rule.

Polynomial Integration

When dealing with polynomial integration, you're often required to integrate terms where \(x\) is raised to some power. This is where polynomial integration shines because it is quite straightforward compared to other integration methods.For successful polynomial integration, follow these steps:

• Rewrite all terms so they are expressed as powers of \(x\). This is crucial as it enables straightforward application of the power rule.
• Identify each component of the polynomial separately. In our example, after distributing \(\sqrt{x}\), the polynomial became \(x^{1/2}\) and \(x^{7/2}\).

Once you've rewritten these terms according to their powers, utilize the power rule for integration on each term individually. It reduces a potentially overwhelming problem into smaller, more manageable tasks, which you then combine at the end.Polynomial integration is great for solidifying understanding of integration because of its repetitive yet simple application of the power rule. When done correctly, you compile these individual results back into the original expression, ensuring you don't forget the constant of integration, \(C\), which is always part of indefinite integrals.