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Chapter 8: Problem 21

Finding an Indefinite Integral In Exercises \(15-34\) , find the indefinite integral. (Note: Solve by the simplest method- not all require integration by parts.) $$\int \frac{\ln x}{x^{3}} d x$$

Short Answer

Expert verified
The indefinite integral of \( \int \frac{\ln x}{x^{3}} dx \) is \( -\frac{\ln x}{2x^{2}} + \frac{1}{4x^{2}} + C \)

Step by step solution

01

Identify u and dv

Generally, we choose \(u\) to be a function which becomes simpler after differentiating and \(dv\) to be a function of which we can easily find the integral. So, we will choose \(u = \ln x\) and \(dv = 1/x^{3} dx\). Therefore, \(du = 1/x dx\) and \(v = -1/(2x^{2})\)
02

Apply the integration by parts formula

Insert the identified \(u\), \(v\), \(du\), and \(dv\) into the formula of integration by parts to solve for the integral. The formula is \( 鈭玼dv = uv - 鈭玽 du \). When we plug the values, this then becomes \( \int \frac{\ln x}{x^{3}} dx = u*v - \int v * du = -\frac{\ln x}{2x^{2}} - \int -\frac{1}{2x^{3}} dx \)
03

Evaluate the integral

Solve the remaining integral of \( \int -\frac{1}{2x^{3}} dx \) by using the power rule of integral. The power rule states that \( 鈭玿^n dx = \frac{x^{n+1}}{n+1} + C \). At this stage, we get \( \int \frac{\ln x}{x^{3}} dx = -\frac{\ln x}{2x^{2}} + \frac{1}{4x^{2}} + C \)

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Most popular questions from this chapter

True or False? In Exercises 81-86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$\begin{array}{l}{\text { If } f \text { is continuous on }[0, \infty) \text { and } \lim _{x \rightarrow \infty} f(x)=0, \text { then } \int_{0}^{\infty} f(x) d x} \\ {\text { converges. }}\end{array}$$

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. $$\int_{2}^{4} \frac{1}{(x-1)^{2}} d x$$

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

Proof Prove that Simpson's Rule is exact when approximating the integral of a cubic polynomial function, and demonstrate the result with \(n=4\) for \(\int_{0}^{1} x^{3} d x\)

Finding or Evaluating an Integral In Exercises \(55-62\) find or evaluate the integral. $$\int \frac{4}{\csc \theta-\cot \theta} d \theta$$
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