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Chapter 28: Problem 23

In Exercises \(3-26,\) integrate each of the given functions. $$\int(x+4)^{8}(x+5) d x$$

Short Answer

Expert verified
The integral is \( \frac{(x+4)^{10}}{10} + \frac{(x+4)^9}{9} + C \).

Step by step solution

01

Identify Parts for Substitution

Recognize that the integrand is the product of two functions, \( (x+4)^8 \text{ and } (x+5)\). We can use substitution to simplify this. Let's set \( u = x+4 \) and thus \( du = dx\). Observe that \((x+5) = (u+1)\).
02

Substitute and Adjust Limits

After substitution, the integral becomes \( \int u^8 (u+1) \, du \). This step simplifies the product of the functions before integration.
03

Expand the Integrand

Expand the integrand \( u^8(u+1) = u^9 + u^8 \). This makes the integral separable, allowing us to integrate term by term.
04

Integrate Term by Term

Integrate each term separately: \( \int u^9 \, du = \frac{u^{10}}{10} \)and\( \int u^8 \, du = \frac{u^9}{9} \).
05

Combine the Integrals

Combine the separate integrals to form: \( \frac{u^{10}}{10} + \frac{u^9}{9} \).
06

Substitute Back to Original Variable

Revert the substitution \( u = x+4 \) back into the function: \( \frac{(x+4)^{10}}{10} + \frac{(x+4)^9}{9} + C \), where \( C \) is the arbitrary constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution Method
The Substitution Method is a powerful tool used in integration to simplify complex integrals by changing variables. It often makes an integral easier to solve by reducing it to a more recognisable form. In the given problem, we are dealing with the integrand
  • \((x+4)^8 (x+5)\)
To use substitution, notice useful patterns or substitutions. Here, setting
  • \(u = x+4\)
turns the expression
  • \((x+5)\) into \((u+1)\).
This makes the variables easier to manage, turning the original integral into
  • \(\int u^8(u+1) du\).
This substitution simplifies the process, helping you to focus on integrating simpler polynomial expressions.
Polynomial Integration
Polynomial integration involves integrating functions of the form
  • \(ax^n\)
where
  • \(a\) is a constant and \(n\) is an integer.
The formula for integrating such terms is:
  • \(\int ax^n dx = \frac{ax^{n+1}}{n+1} + C\),
where
  • \(C\) is the constant of integration.
For our problem, once substitution is made, expand
  • \(u^8(u+1)\) to \(u^9 + u^8\).
This allows us to integrate term by term:
  • \(\int u^9 \, du\) results in \(\frac{u^{10}}{10}\)
  • \(\int u^8 \, du\) results in \(\frac{u^9}{9}\).
These separate integrals can be added together to achieve the integrated expression in terms of
  • \(u\).
Definite and Indefinite Integrals
Integrals are classified into two main types: definite and indefinite. In indefinite integrals, as seen in our exercise, the integral does not have bounds. This generally results in a family of functions plus a constant
  • \(C\).
In the definite integral, you compute the actual value with specific limits, but since our example focuses on indefinite integration, remember: the output is another function, not a static figure.
  • Thus, \(\int u^9 du\) results in \(\frac{u^{10}}{10} + C\).
An essential aspect of indefinite integrals is this constant
  • \(C\), which represents an unknown constant that could arise from any constant term added during differentiation.
Step-by-Step Solutions
Step-by-step solutions are incredibly valuable, particularly as they break down complex problems into smaller, more manageable parts. This method acts as a roadmap, guiding you through the integration process clearly and sequentially. In our exercise, we:
  • Identified the parts for substitution, setting \(u = x+4\), and noticed \((x+5) = (u+1)\).
  • Substituted these into the integral, simplifying our problem.
  • Expanded the integrand, allowing polynomial integration of each term separately.
  • Integrated each term one by one, using the rules of polynomial integration.
  • Finally combined the results and reverted the substitution, converting back to the original variable \(x\).
This step-by-step breakdown improves understanding, allowing for easier tracking of each transformation from start to finish.

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