Problem 37 - Essential Calculus

Question

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{1}^{e} \frac{(1+\ln x)^{2}}{x} d x $$

Step-by-Step Solution

Answer

The result of the integral is \(7/3\).

Step 1: Analyze the form of the integral

The present integral is not in a standard form that allows us to immediately guess the antiderivative or perform a straightforward substitution. However, by recognizing the composite function structure in the numerator, we see an opportunity for the substitution method, where \(u = 1+ \ln x \). To use this substitution, we need to express everything in the integral in terms of \( u \), including the limits of integration and \( dx \).

Step 2: Perform the substitution

Substitute \(u = 1+ \ln x\). To change \( dx \) to \( du \), we differentiate both sides of \(u = 1+\ln x\) with respect to \(x\), yielding \[ du/dx = 1/x \]. Therefore, \( dx = x du \). Substitute \( dx \) with \(xdu\) in the integral. The limits of integration must also be changed from \[ x = 1 \rightarrow u = 1+ \ln 1 = 1 \] and \[ x=e \rightarrow u= 1+ \ln e = 2 \]. The integral now becomes \[ \int_{1}^{2}u^{2} du \].

Step 3: Evaluate the integral

The antiderivative of \( u^2 \) with respect to \( u \) is \( u^3/3 \). Substituting this into the integral, we get \[ [ u^3/3 ]_{1}^{2} = 2^3/3 - 1^3/3 = 8/3 - 1/3 = 7/3 \].

Step 4: Verify Result with a Graphing Utility

Finally, one would use a graphing software or calculator to plot the original function and calculate the area under the curve from \( x = 1 \) to \( x = e \). The calculated area should confirm the analytical solution obtained.