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Chapter 8: Problem 21

Use integration tables to find the indefinite integral. \(\int \frac{4 x}{(2-5 x)^{2}} d x\)

Short Answer

Expert verified
The indefinite integral of \(\int \frac{4 x}{(2-5 x)^{2}} d x\) is \(\frac{4}{5(-2+5x)}+C\).

Step by step solution

01

Transformation of the Integral

We start by rewriting the integrand to \(\int -\frac{4x}{(-2+5x)^{2}} d x\). A resemblance to one of the standard integrals in the integral table: \(\int \frac{u'}{u^2}\) can be seen, with \(u = -2 + 5x\) and \(u' = 5 dx\).
02

Implementing the Standard Integral

Replace \(u = -2 + 5x\) and \(u' = 5 dx\) in the integral, which now reads \(-\frac{4}{5}\int \frac{u'}{u^{2}}\). Using the standard integral \(\int \frac{u'}{u^{2}} du = -\frac{1}{u}\), this becomes \(-\frac{4}{5} \left(-\frac{1}{u}\right) = \frac{4}{5u}\).
03

Substitute for 'u' and Add Constant of Integration

Replacing 'u' with original term gives the answer \(\frac{4}{5(-2+5x)}\). Don't forget to add the constant of integration 'C' to the answer since the integral was indefinite. The final answer is \(\frac{4}{5(-2+5x)}+C\).

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Key Concepts

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

Integration Tables
Integration tables are essential tools in calculus to help you integrate various functions quickly and easily. These tables provide ready-made solutions for common forms of integrals so you can avoid getting bogged down with complex calculations.
Such tables list integrals like
  • \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \)
  • \( \int e^x dx = e^x + C \)
  • \( \int \frac{1}{ax+b} dx = \frac{1}{a}\ln|ax+b| + C \)
By using these pre-determined solutions, you essentially match your integral problem to an entry in the table. This can speed up the process significantly.
For the exercise, noticing how the problem fits into a known form like \( \int \frac{u'}{u^2} \) from the table allows for straightforward integration using substitutions.
Standard Integrals
Standard integrals are foundational pieces of calculus that represent integrals of basic functions. Usually, they are listed in integration tables and form the basis for more complex integrals through transformations and substitutions.
  • For instance, \( \int \frac{u'}{u^2} du = -\frac{1}{u} + C \) is a standard form that makes life easier when integrating fractions.
  • These standards simplify a broad range of integrals, serving as basic building blocks.
When you identify a complex integral as having a similar structure to a standard one, you can employ the standard solution immediately.
This simplifies the problem-solving process tremendously, as seen in the exercise where the problem was transformed into a standard integral form before applying the solution.
Transformation of Integrals
Transforming integrals involves rewriting the original problem into a form that is easier to integrate. This often includes making substitutions or rearranging the expression to match a standard integral.
  • This technique allows us to simplify the function we're working with.
  • We can make the function more recognizable and manageable.
For example, in the original exercise, the substitution \( u = -2 + 5x \) was used to convert the integral into a form that resembles \( \int \frac{u'}{u^2} du \).
By transforming it this way, you can then apply a standard solution directly, which is a powerful method to tackle seemingly complex integrals.
Thus, transformation is a key step before directly applying integration formulas from tables, and greatly simplifies the process of finding indefinite integrals.

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Most popular questions from this chapter

Indeterminate Forms Show that the indeterminate forms \(0^{0}, \infty^{0},\) and \(1^{\infty}\) do not always have a value of 1 by evaluating each limit. (a) \(\lim _{x \rightarrow 0^{+}} x^{\ln 2 /(1+\ln x)}\) (b) \(\lim _{x \rightarrow \infty} x^{\ln 2 /(1+\ln x)}\) (c) \(\lim _{x \rightarrow 0}(x+1)^{(\ln 2) / x}\)

Evaluating an Improper Integral In Exercises \(33-48\) 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. $$ \int_{0}^{\pi / 2} \tan \theta d \theta $$

Evaluate \(\lim _{x \rightarrow \infty}\left[\frac{1}{x} \cdot \frac{a^{x}-1}{a-1}\right]^{1 / x}\) where \(a>0, a \neq 1\)

Evaluating an Improper Integral In Exercises \(33-48\) 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. $$ \int_{0}^{1} x \ln x d x $$

Evaluating an Improper Integral In Exercises \(33-48\) 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. $$ \int_{0}^{5} \frac{1}{25-x^{2}} d x $$
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