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Chapter 8: Problem 39

$$\text {Evaluate the following integrals.}$$ $$\int \frac{81}{x^{3}-9 x^{2}} d x$$

Short Answer

Expert verified
Question: Evaluate the integral: $$\int\frac{81}{x^3 - 9x^2}\,dx$$ Answer: $$\int\frac{81}{x^3 - 9x^2}\,dx = \ln\lvert x(x-9)^2 \rvert + C$$

Step by step solution

01

Identifying $$f(x)$$ and $$f'(x)$$

To perform natural logarithmic integration we need to have the integral in the form $$\int\frac{f'(x)}{f(x)}\,dx$$. We can rewrite the given integral as $$\int\frac{9(9 - x)(1)}{x(x-9)^2}\,dx$$, so we can identify the following: $$f(x) = x(x-9)^2$$ Now, we must differentiate $$f(x)$$ to find $$f'(x)$$: $$f'(x) = \frac{d}{dx}\left[x(x-9)^2\right] = (x-9)^2 + 2x(x-9) = 9(9 - x)$$
02

Applying Natural Logarithmic Integration

Since we have the integral in the desired form, we can apply the natural logarithmic integration to find the solution: $$\int\frac{f'(x)}{f(x)}\,dx = \ln\lvert f(x)\rvert + C$$ Substituting $$f(x)$$: $$\int\frac{81}{x^3 - 9x^2}\,dx = \ln\lvert x(x-9)^2 \rvert + C$$
03

Expressing the Final Answer

Now that we have found the integral, we can write down the final answer with the constant of integration: $$\int\frac{81}{x^3 - 9x^2}\,dx = \ln\lvert x(x-9)^2 \rvert + C$$

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Key Concepts

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

Natural Logarithmic Integration
Natural logarithmic integration revolves around integrating functions of the form \( \int \frac{f'(x)}{f(x)}\,dx \). This method is particularly useful because it simplifies the integration process into a logarithmic form. Breaking it down, you look for an integrand that can be separated into the derivative numerator \( f'(x) \), and the function denominator \( f(x) \).

Here's how it works:
  • Find the function \( f(x) \) such that when differentiated, the result is involved in the numerator of the integrand.
  • Delineate the derivative form, \( f'(x) \), and ensure it can be paired with \( f(x) \) appropriately.
  • The integration is concluded by integrating the form \( \int\frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C \).
In our exercise, this took the form of identifying \( f(x) = x(x-9)^2 \) which was differentiated to yield the corresponding \( f'(x) \), facilitating natural logarithm integration.
Differentiation
Differentiation is the process of finding the derivative of a function. It reveals the rate at which a particular quantity changes. The derivative itself is foundational to calculus, paving the way for numerous applications, including integration by parts and natural logarithmic integration.

When differentiating products of terms, as seen in our example \( x(x-9)^2 \), the product rule is applied:
  • For two functions \( u(x) \) and \( v(x) \), the derivative is \( u'(x)v(x) + u(x)v'(x) \).
  • Thus, for \( x(x-9)^2 \): different parts are differentiated and then combined.
  • The derivative \( f'(x) = (x-9)^2 + 2x(x-9) \) was expanded and simplified to represent \( 9(9 - x) \).
Differentiation in this manner sets the stage for employing other calculus techniques like natural logarithmic integration.
Constant of Integration
The constant of integration, denoted by \( C \), is an integral part of integral calculus. It's crucial for expressing indefinite integrals, ensuring the solution encompasses all potential antiderivatives of the function.

Here's why it's needed:
  • When integrating, the process reverses differentiation which "loses" any constant that might have been present since the derivative of a constant is zero.
  • By including \( C \), we account for any possible initial values or shifts in the function, offering a family of solutions rather than a single one.
  • In our problem, the natural logarithmic integration concluded with \( \ln|f(x)| + C \), capturing the general solution set.
Remembering to include \( C \) helps secure the mathematical integrity of the infinite solutions present in indefinite integrals.

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Most popular questions from this chapter

It can be shown that $$\begin{array}{l}\int_{0}^{\pi / 2} \sin ^{n} x d x=\int_{0}^{\pi / 2} \cos ^{n} x d x= \\\\\quad\left\\{\begin{array}{ll}\frac{1 \cdot 3 \cdot 5 \cdot \cdots(n-1)}{2 \cdot 4 \cdot 6 \cdots n} \cdot \frac{\pi}{2} & \text { if } n \geq 2 \text { is an eveninteger } \\\\\frac{2 \cdot 4 \cdot 6 \cdots(n-1)}{3 \cdot 5 \cdot 7 \cdots n} & \text { if } n \geq 3 \text { is an odd integer. }\end{array}\right.\end{array}$$ a. Use a computer algebra system to confirm this result for \(n=2,3,4,\) and 5 b. Evaluate the integrals with \(n=10\) and confirm the result. c. Using graphing and/or symbolic computation, determine whether the values of the integrals increase or decrease as \(n\) increases.

The family \(f(x)=\frac{1}{x^{p}}\) revisited Consider the family of functions \(f(x)=\frac{1}{x^{p}},\) where \(p\) is a real number. For what values of \(p\) does the integral \(\int_{0}^{1} f(x) d x\) exist? What is its value?

Evaluate the following integrals. $$\int_{-2}^{-1} \sqrt{-4 x-x^{2}} d x$$

Let \(a>0\) and let \(R\) be the region bounded by the graph of \(y=e^{-a x}\) and the \(x\) -axis on the interval \([b, \infty)\). a. Find \(A(a, b),\) the area of \(R\) as a function of \(a\) and \(b\). b. Find the relationship \(b=g(a)\) such that \(A(a, b)=2\). c. What is the minimum value of \(b\) (call it \(b^{*}\) ) such that when \(b>b^{*}, A(a, b)=2\) for some value of \(a>0 ?\)

Evaluate \(\int_{0}^{\pi / 4} \ln (1+\tan x) d x\) using the following steps. a. If \(f\) is integrable on \([0, b],\) use substitution to show that $$\int_{0}^{b} f(x) d x=\int_{0}^{b / 2}(f(x)+f(b-x)) d x$$ b. Use part (a) and the identity tan \((\alpha+\beta)=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta}\) to evaluate \(\int_{0}^{\pi / 4} \ln (1+\tan x) d x\) (Source: The College Mathematics Journal, \(33,4,\) Sep 2004 )
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