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Chapter 8: Problem 18

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \frac{1}{x(\ln x)^{3}} d x $$

Short Answer

Expert verified
The integral evaluates to \( -\frac{1}{2(\ln x)^2} + C \)

Step by step solution

01

Substitution

Let \( u = \ln x \). Therefore, as \( u = \ln x \), differentiating both sides with respect to \( x \) gives \( du = \frac{1}{x} dx \). Thus, \( dx = x du \).
02

Re-write Integral

Substitute \( u = \ln x \) and \( dx = x du \) into the integral. This transforms the original integral to \( \int \frac {1}{u^3} du \).
03

Integrate

Evaluate the integral \( \int \frac {1}{u^3} du \). The resulting integral is implied by the power rule redux for integration. Thus, the integral evaluates to \( -\frac{1}{2u^2} \).
04

Back substitution

Substitute \( u = \ln x \) back into the result from step 3 in order to express this in terms of the original variable, \( x \). Our final solution: \( -\frac{1}{2(\ln x)^2} + C \), where \( C \) is a constant of integration.

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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 in integral calculus. It simplifies the integration process. When using this method, our goal is to replace a part of the integral with a single variable, which makes the integration easier.
Here's how it works:
  • First, identify a part of the integrand that can be set as a new variable. This is typically something complicated or something appearing in a denominator.
  • Perform a substitution. In our example, we set \( u = \ln x \). This simplifies our problem by removing the natural logarithm.
  • Next, we find the differential of this new variable. That is, differentiate \( u = \ln x \) with respect to \( x \) to find \( du = \frac{1}{x} dx \).
Once the substitution and differentiation are complete, rewrite the entire integral in terms of the new variable. Our original integral \( \frac{1}{x(\ln x)^3} dx \) transforms to \( \int \frac{1}{u^3} du \). This method often turns a difficult problem into a much simpler one.
Natural Logarithm
The natural logarithm, denoted as \( \ln x \), is the logarithm to the base \( e \), where \( e \) is approximately 2.71828. This function is crucial in calculus, especially when dealing with exponential growth and decay.
Here are a few important properties:
  • \( \ln(e^x) = x \) which reflects the inverse relationship between the natural logarithm and the exponential function.
  • The derivative of \( \ln x \) is \( \frac{1}{x} \). This property is essential and used in finding differentials during substitution.
In our integration example, setting \( \ln x = u \) allowed us to use these properties effectively.
Understanding the behavior of \( \ln x \), particularly its role in reversing exponential functions, is key when working through integration problems.
Integral Calculus
Integral calculus focuses on finding the anti-derivatives of functions. This process is often about reversing the differentiation. There are various techniques for solving integrals, among which substitution is just one.
Here's a rundown of solving integrals using substitution:
  • First, use the substitution to convert the integral into terms of a new variable.
  • Solve the integral, as we did with \( \int \frac{1}{u^3} du \). Applying the power rule gives us \( -\frac{1}{2u^2} \).
  • Lastly, substitute back the original variable to express the solution in its initial terms.
The constant \( C \) signifies that there are infinitely many antiderivatives of a function, differing by a constant. This constant is essential when talking about indefinite integrals. Thus, our solution becomes \( -\frac{1}{2(\ln x)^2} + C \).
Mastering integral calculus involves knowing when to apply each technique, which is a key part of understanding and solving problems.

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Most popular questions from this chapter

Use integration by parts to verify the reduction formula. \(\int \cos ^{n} x d x=\frac{\cos ^{n-1} x \sin x}{n}+\frac{n-1}{n} \int \cos ^{n-2} x d x\)

Volume The region bounded by \(y=e^{-x^{2}}, y=0, x=0\), and \(x=b(b>0)\) is revolved about the \(y\) -axis. (a) Find the volume of the solid generated if \(b=1\). (b) Find \(b\) such that the volume of the generated solid is \(\frac{4}{3}\) cubic units.

Prove the following generalization of the Mean Value Theorem. If \(f\) is twice differentiable on the closed interval \([a, b]\), then $$ f(b)-f(a)=f^{\prime}(a)(b-a)-\int_{a}^{b} f^{\prime \prime}(t)(t-b) d t $$

State (if possible) the method or integration formula you would use to find the antiderivative. Explain why you chose that method or formula. Do not integrate. $$ \int e^{2 x} \sqrt{e^{2 x}+1} d x $$

Rewrite the improper integral as a proper integral using the given \(u\) -substitution. Then use the Trapezoidal Rule with \(n=5\) to approximate the integral. $$ \int_{0}^{1} \frac{\cos x}{\sqrt{1-x}} d x, \quad u=\sqrt{1-x} $$
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