Question

What equation results if integration by parts is applied to the integral \(\int \frac{1}{x \ln x} d x\) with the choices \(\mathrm{u}=\frac{1}{\ln \mathrm{x}}\) and \(\mathrm{dv}=\frac{1}{\mathrm{x}} \mathrm{d} \mathrm{x}\) ? In what sense is this equation true? In what sense is it false ?

Short answer

Is the equation true or false? Answer: The resulting equation is \(\int \frac{1}{x \ln x} dx = \frac{\ln|x|}{\ln x} - \int \frac{-\ln|x|}{x(\ln x)^2}dx - \int \frac{-C}{x(\ln x)^2}dx\). The equation is true as it correctly follows integration by parts, but it is false in that it does not make the integral simpler to solve, which is the main goal of applying integration by parts.

Step-by-step solution

Find du

Given: \(u = \frac{1}{\ln x}\), so differentiate \(u\) with respect to \(x\) to find \(du\). \(du = -\frac{1}{(\ln x)^2}\frac{d(\ln x)}{dx} dx = -\frac{1}{x(\ln x)^2}dx\)

Find v

Given: \(dv = \frac{1}{x} dx\), so integrate \(dv\) with respect to \(x\) to find \(v\). \(v = \int \frac{1}{x} dx = \ln|x| + C\)

Apply the integration by parts formula

Now we will apply the integration by parts formula: \(\int \frac{1}{x \ln x} dx = u\cdot v - \int v \cdot du\) Substitute the values of \(u, v, du\) that we found in Steps 1 and 2: \(= \left(\frac{1}{\ln x}\right)(\ln|x| + C) - \int \left(\ln|x| + C\right)\cdot \left(-\frac{1}{x(\ln x)^2}\right)dx\)

Evaluate the resulting integral

Simplify and solve the remaining integral: \(\int \frac{1}{x \ln x} dx = \frac{\ln|x|}{\ln x} - \int \frac{-\ln|x|}{x(\ln x)^2}dx - \int \frac{-C}{x(\ln x)^2}dx\) This is the resulting equation when integration by parts is applied to the given integral with the specified choices for \(u\) and \(dv\). As for the validity of the equation, it is true in the sense that it follows the integration by parts procedure correctly. However, it is false in that it leads to an integral that is not simpler to evaluate than the original one. Integration by parts is typically used to make an integral more straightforward or solvable, but in this case, it does not achieve that goal.