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Chapter 8: Problem 23

Finding an Indefinite Integral In Exercises \(19-40\) , use a table of integrals to find the indefinite integral. $$\int \frac{x}{(7-6 x)^{2}} d x$$

Short Answer

Expert verified
\(\frac{1}{3} (7-6x)^{-1} + C.\

Step by step solution

01

Title: Identification

We identify the integral \(\int \frac{x}{(7-6 x)^{2}} d x\). The form of this integral suggests using the equation \(\int \frac{1}{x^{n}} d x = \frac{1}{n-1} x^{n-1} + C\) where \(n \neq 1\), with the appropriate substitution.
02

Title: Substitution

Let's make a substitution to simplify the integral. We let \(u = 7 - 6x\), thus \(du = -6dx\). This means \(dx = -\frac{1}{6} du\). Also, notice that \(x = \frac{7-u}{6}\). So, after substitution, we have the integral as -\frac{1}{6} \int u^{-2} du.
03

Title: Apply the Integration Rule

Now we apply the rule \(\int x^{n} dx = \frac{1}{n+1} x^{n+1} + C\) where \(n \neq -1\). When applied, we get -\frac{1}{6} * -2u^{-1} + C = \frac{1}{3} u^{-1} + C.
04

Title: Resubstituting \(u\)

Now, replace \(u\) with \(7-6x\) to get the answer in terms of \(x\). Hence, we get \(\frac{1}{3} (7-6x)^{-1} + C\).

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Key Concepts

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

Integration by Substitution
Integration by substitution is a technique used to simplify integrals, making them easier to solve. It involves replacing a difficult part of the integral with a new variable, often denoted by \(u\). In this exercise, we let \(u = 7 - 6x\). This substitution simplifies the function inside the integral.

Here's how it works:
  • Choose a substitution that transforms a difficult part of the integral into a simpler form.
  • Express \(dx\) in terms of \(du\) by differentiating \(u\) with respect to \(x\). This gives \(du = -6dx\) which rearranges to \(dx = -\frac{1}{6}du\).
  • Substitute both \(u\) and \(dx\) back into the integral to transform it, often making the integral easier to handle.
This method is particularly effective when a function and its derivative appear within the integral, as it allows you to directly replace complex expressions with simpler variables. Ultimately, integration by substitution reduces the complexity of the original problem.
Table of Integrals
A table of integrals is a handy tool in calculus, providing a list of commonly encountered integrals along with their solutions. It's like a directory for integration problems, and can be particularly useful when a specific integral form matches one from the table.

In our exercise, we initially transformed the integral to a form that could be cross-referenced with such tables. The process is as follows:
  • Identify the transformed integral and find a matching format in the integral table.
  • Use the provided integral solution from the table to solve the related integral.
Tables of integrals are especially helpful when dealing with difficult integrals, allowing you to bypass lengthy calculations and directly apply known solutions. This saves time and simplifies the task without always requiring the derivation from scratch.
Integration Rules
Integration rules are the basic guidelines and formulas used to perform integration, similar to rules in differentiation. They help determine the anti-derivative for many different types of functions.

In the exercise solution, once we had simplified the integral using substitution, we used an integration rule for powers of \(x\). Specifically:
  • The rule \(\int x^n \, dx = \frac{1}{n+1} x^{n+1} + C\) was used after substitution resulted in an integrand fitting this form.
  • We applied this rule to \(\int u^{-2} \, du\) to find \(-\frac{1}{6} \times \int u^{-2} \, du = \frac{1}{3} u^{-1} + C\).
These integration rules are crucial in systematically solving integrals. They offer a straightforward path to finding antiderivatives and play a significant role in calculus education and application.

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Most popular questions from this chapter

Integration by Tables In Exercises 13 and \(14,\) use a table of integrals with forms involving ln \(u\) to find the indefinite integral. $$\int x^{7} \ln x d x$$

In Exercises 91-98, find the Laplace Transform of the function. $$f(t)=\sinh a t$$

Estimating Errors \(\quad\) In Exercises \(25-28\) , use the error formulas in Theorem 8.6 to estimate the errors in approximating the integral, with \(n=4,\) using \((a)\) the Trapezoidal Rule and (b) Simpson's Rule. $$\int_{0}^{2}\left(x^{2}+2 x\right) d x$$

\(33-48,\) determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$\int_{4}^{\infty} \frac{\sqrt{x^{2}-16}}{x^{2}} d x$$

Approximating a Function The table lists several measurements gathered in an experiment to approximate an unknown continuous function \(y=f(x)\) . \(\begin{array}{|c|c|c|c|c|c|}\hline x & {0.00} & {0.25} & {0.50} & {0.75} & {1.00} \\ \hline y & {4.32} & {4.36} & {4.58} & {5.79} & {6.14} \\\ \hline\end{array}\) \(\begin{array}{|c|c|c|c|c|}\hline x & {1.25} & {1.50} & {1.75} & {2.00} \\\ \hline y & {7.25} & {7.64} & {8.08} & {8.14} \\ \hline\end{array}\) (a) Approximate the integral \(\int_{0}^{2} f(x) d x\) using the Trapezoidal Rule and Simpson's Rule. (b) Use a graphing utility to find a model of the form \(y=a x^{3}+b x^{2}+c x+d\) for the data. Integrate the resulting polynomial over \([0,2]\) and compare the result with the integral from part (a).
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