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

\( \frac{2x^{9/2}}{9} + \frac{7x^{9/7}}{9} + C \)

Step 1: Identify the Integral

The problem requires finding the indefinite integral of the function \( f(x) = x^{7/2} + x^{2/7} \). First, write down the integral to be solved: \[ \int \left( x^{7/2} + x^{2/7} \right) dx. \] This integral can be broken down into two simpler integrals to find separately.

Step 2: Apply the Power Rule for Integration

The power rule for integration states that for any real number \( n eq -1 \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \). This rule is applicable to each term of our integral: \( \int x^{7/2} \, dx \) and \( \int x^{2/7} \, dx \).

Step 3: Integrate the First Term

Using the power rule on the first term \( x^{7/2} \), we get: \[ \int x^{7/2} \, dx = \frac{x^{7/2 + 1}}{7/2 + 1} = \frac{x^{9/2}}{9/2}. \] This simplifies to: \[ \frac{2x^{9/2}}{9}. \]

Step 4: Integrate the Second Term

Now apply the power rule to the second term \( x^{2/7} \): \[ \int x^{2/7} \, dx = \frac{x^{2/7 + 1}}{2/7 + 1} = \frac{x^{9/7}}{9/7}. \] This simplifies to: \[ \frac{7x^{9/7}}{9}. \]

Step 5: Combine the Results

Combine the results of the two separate integrals to get the final solution: \[ \int \left(x^{7/2} + x^{2/7}\right) dx = \frac{2x^{9/2}}{9} + \frac{7x^{9/7}}{9} + C, \] where \( C \) is the constant of integration.

Key Concepts

Power Rule for Integration

To grasp the power rule for integration, think of it as a handy tool for finding the antiderivative of polynomials. When you want to find the integral of a function that involves a power of a variable, this rule is your best friend. The power rule states: for any real number \( n eq -1 \), the indefinite integral can be calculated as \( \int x^n \ dx = \frac{x^{n+1}}{n+1} + C \). Here are some key points about this rule:

• The exponent \( n \) in \( x^n \) is increased by one to get \( n+1 \).
• You divide by the new exponent, \( n+1 \), giving you \( \frac{x^{n+1}}{n+1} \).
• "\( C \)" is the constant of integration, a vital part of the result to indicate any constant can be added to the antiderivative.

Using this rule makes integrating powers of \( x \) straightforward and systematic, especially when dealing with polynomial expressions.

Integration of Polynomials

Polynomials are expressions made up of terms called monomials. Integrating a polynomial means finding the antiderivative of each term individually, and then adding the results together. The great news is that the power rule for integration directly applies to each polynomial term. Consider the polynomial \( x^{a/b} + x^{c/d} \). To integrate, perform these steps for each term:

• Apply the power rule to each term separately, as shown in the step-by-step solution for \( \int x^{7/2} \ dx \) and \( \int x^{2/7} \ dx \).
• Simplify the expression to make calculation easier, turning fractions into simpler forms.

After you integrate each term, sum the results. This integration method can address any polynomial combination, ensuring you fully find the indefinite integral for the entire polynomial expression.

Constant of Integration

The constant of integration, represented by \( C \), is an essential part of indefinite integrals. This constant is vital because indefinite integrals yield a family of functions, differing by a constant. Here’s why \( C \) is crucial:

• When you take the derivative of any constant added to a function, it vanishes; hence any constant can accompany the antiderivative.
• Including \( C \), the solutions account for all possible original functions that could have led to the given derivative.

In the context of integration, always remember that various functions can produce the same derivative apart from different constant values. Thus, each indefinite integral solution should have \( C \) to be complete and correct.