Problem 49 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int\left(x^{7 / 2}+x^{2 / 7}\right) d x $$

Step-by-Step Solution

Answer

\( \int \left(x^{7/2} + x^{2/7}\right) dx = \frac{2}{9}x^{9/2} + \frac{7}{9}x^{9/7} + C \)

Step 1: Identify the Integral Components

The given problem is to compute the indefinite integral \( \int \left(x^{7/2} + x^{2/7}\right) dx \). Identify each term separately: \( x^{7/2} \) and \( x^{2/7} \) are the components that need to be integrated individually.

Step 2: Apply the Power Rule for Integrals

Utilize the power rule for integrals: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \). Apply this to each term separately. For the first term, \( x^{7/2} \), and for the second term, \( x^{2/7} \).

Step 3: Integrate the First Term

Using the power rule for the first term, \(x^{7/2}\):\[\int x^{7/2} \, dx = \frac{x^{7/2 + 1}}{7/2 + 1} = \frac{x^{9/2}}{9/2} = \frac{2}{9}x^{9/2}\]

Step 4: Integrate the Second Term

Using the power rule for the second term, \(x^{2/7}\):\[\int x^{2/7} \, dx = \frac{x^{2/7 + 1}}{2/7 + 1} = \frac{x^{9/7}}{9/7} = \frac{7}{9}x^{9/7}\]

Step 5: Combine the Results

Combine the integrals of both terms to find the solution of the indefinite integral:\[\int \left(x^{7/2} + x^{2/7}\right) dx = \frac{2}{9}x^{9/2} + \frac{7}{9}x^{9/7} + C\]where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integrals

The power rule for integrals is one of the most fundamental tools in calculus for finding indefinite integrals. It states that for any real number \( n \) (except \( n = -1 \)), the integral of \( x^n \) with respect to \( x \) is given by:\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]where \( C \) is the constant of integration. This rule helps simplify the process of integrating polynomial expressions.
In our specific problem, we apply this rule to each term of the integrand separately, namely \( x^{7/2} \) and \( x^{2/7} \). By using the power rule, we transform the process of integration into a straightforward algebraic manipulation. This makes it easier to handle even fractional exponents as seen in this example.

Integration Techniques

When dealing with the integration of expressions, it's important to apply appropriate techniques to solve them effectively. The power rule is one technique, especially useful for polynomials or terms with powers.For our problem, after identifying the terms \( x^{7/2} \) and \( x^{2/7} \), we utilized this technique separately for each term:

• First, for \( x^{7/2} \), adding 1 gives \( x^{9/2} \). This is then divided by \( 9/2 \), resulting in \( \frac{2}{9}x^{9/2} \).
• Second, the term \( x^{2/7} \) became \( x^{9/7} \) after adding 1. Dividing by \( 9/7 \) yields \( \frac{7}{9}x^{9/7} \).

Combining Results

Once each term is integrated separately, they are combined to form the final solution. This often creates a more manageable way to handle integrals, as the process can be broken into simpler parts.

Constant of Integration

The constant of integration, often denoted as \( C \), is a crucial component in the solution to an indefinite integral. This constant represents the family of all possible antiderivatives of a function.When taking indefinite integrals, we acknowledge that differentiating any constant results in zero. Thus, adding \( C \) incorporates all vertical shifts of the antiderivative.In the context of our exercise, after integrating both terms, we include \( C \) in the final result:\[\int \left(x^{7/2} + x^{2/7}\right) dx = \frac{2}{9}x^{9/2} + \frac{7}{9}x^{9/7} + C\]This ensures we cover all potential solutions, allowing flexibility in interpretation and application of the integral in different scenarios. Remember, though the value of \( C \) doesn't change the overall shape of the function's graph, it shifts it vertically which is crucial for specific applications.