Problem 31 - Biocalculus Calculus for the Life Sciences

Question

Evaluate the indefinite integral. \(\int x(2 x+5)^{8} d x\)

Step-by-Step Solution

Answer

\(\frac{1}{40} (2x + 5)^{10} - \frac{5}{36} (2x + 5)^9 + C\)

Step 1: Recognize the Structure

Notice that the integrand is in the form of a product of a function and its derivative. Here, the function inside the integral is \((2x + 5)^8\), and its derivative is a constant multiple of \(x\). This suggests a substitution method for integration.

Step 2: Choose a Substitution

Let \(u = 2x + 5\). Then, compute the derivative \(du/dx = 2\). Rearranging, we get \(du = 2 \, dx\) or \(dx = \frac{1}{2} du\). Also express \(x\) in terms of \(u\):\(x = \frac{u - 5}{2}\).

Step 3: Substitute into the Integral

Replace all \(x\) and \(dx\) in the original integral:\[\int x(2x+5)^8 \, dx = \int \frac{u - 5}{2} u^8 \cdot \frac{1}{2} \, du\]Simplify this to:\[\frac{1}{4} \int (u - 5) u^8 \, du\]

Step 4: Expand the Integrand

Distribute to expand the integrand:\[\frac{1}{4} \int (u^9 - 5u^8) \, du\]

Step 5: Integrate Term by Term

Integrate each term separately:\[\frac{1}{4} \left( \int u^9 \, du - 5 \int u^8 \, du \right)\]This becomes:\[\frac{1}{4} \left( \frac{u^{10}}{10} - 5 \cdot \frac{u^9}{9} \right) + C\]

Step 6: Simplify the Expression

Simplify the integrated result:\[\frac{1}{4} \left( \frac{u^{10}}{10} - \frac{5u^9}{9} \right)\]This can further be simplified to:\[\frac{1}{40} u^{10} - \frac{5}{36} u^9 + C\]

Step 7: Substitute Back in Terms of x

Replace \(u\) with \(2x + 5\) to get the integral in terms of \(x\):\[\frac{1}{40} (2x + 5)^{10} - \frac{5}{36} (2x + 5)^9 + C\]

Step 8: Final Result

The solution to the integral \(\int x(2x+5)^8 \, dx\) is:\[\frac{1}{40} (2x + 5)^{10} - \frac{5}{36} (2x + 5)^9 + C\]

Key Concepts

Substitution Method

In calculus, the substitution method is a popular technique for solving indefinite integrals, especially when you notice a function and its derivative in the integrand. The goal is to make the integration process easier by transforming a complicated integral into a simpler one.
To use substitution effectively:

• Identify a part of the integrand (a function) to be replaced by a new variable, usually denoted as \( u \).
• Derive \( du \) by differentiating \( u \) with respect to \( x \).
• Solve for \( dx \) in terms of \( du \) or vice-versa. This allows you to replace the original variable with the substitute variable in the integral.

Once these substitutions are made, you perform the integral with respect to \( u \) instead of \( x \). After integrating, substitute back the expression for \( u \) in terms of \( x \) to get the answer in its original variable. This method is powerful because it can turn a complex integral into something much more manageable.

Integration Techniques

Integration techniques are strategies used to find the integral of a function. Besides substitution, several techniques aid in solving various types of integrals. Some of these techniques include:

• Integration by Parts: This technique is useful when the integrand is a product of two functions, based on the product rule of differentiation.
• Partial Fraction Decomposition: Used primarily for rational expressions, it involves expressing a complicated fraction as a sum of simpler fractions that are easier to integrate.
• Trigonometric Integrals: For integrals involving trigonometric functions, identities and substitutions can simplify the process.

In our specific problem, the substitution method was applied because the structure of the integrand—an expression raised to a power coupled with its derivative—makes it an ideal candidate for substitution. Understanding and selecting the correct technique is crucial for efficiently solving any integral.

Integrand Expansion

Expanding the integrand is a common technique used to simplify the integration process. After making a substitution, you might end up with a product or composition of terms that are difficult to integrate. In such cases, expanding the expression into a sum of simpler terms is beneficial.
Here's how this helps:

• The expansion process involves distributing across terms to transform a difficult expression into a series of simpler expressions that are straightforward to integrate.
• Once expanded, you can integrate each term separately, which simplifies the overall computation.

In the given exercise, after substituting \( u = 2x + 5 \), we had an integrand \( \frac{1}{4} \int (u-5)u^8 \, du \). By expanding this to \( \frac{1}{4} \int (u^9 - 5u^8) \, du \), we broke down the task into integrating \( u^9 \) and \( 5u^8 \) separately. This step significantly simplifies the integration process.