Problem 31 - Essential Calculus

Question

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} x \ln x d x $$

Step-by-Step Solution

Answer

The improper integral \(\int_{0}^{1} x \ln x dx\) is convergent and its value equals \(-\frac{1}{4}\).

Step 1: Determine if the Integral Converges or Diverges

Since the integral is improper at \(x=0\), it can be determined whether it converges or diverges by checking the limit as \(x\) approaches \(0\). In this case, the limit at \(x=0\) of the function \(x \ln x\) is \(0\), meaning the integral is thus convergent. This is due to the fact that the natural logarithm function grows more slowly than any power of \(x\).

Step 2: Integration by Parts

The integration can be solved by using the method of integration by parts, with \(u=\ln x\) and \(dv=xdx\). Derived from this, \(du=\frac{1}{x}dx\) and \(v=\frac{1}{2}x^2\). The formula for integration by parts is \(\int udv = uv - \int vdu\). Substituting \(u\), \(v\), \(du\), and \(dv\) will provide the integral evaluation.

Step 3: Solve the Remaining Integral

Applying these to the integration by parts formula, we obtain \(\frac{1}{2}x^2 \ln x - \int \frac{1}{2}x dx\). The second integral is straightforward, \(\frac{1}{4}x^2\). Substituting the limits of \(0\) and \(1\) yields the value of the original integral.

Step 4: Evaluate the Integral

Evaluating from \(0\) to \(1\), we get : \( [\frac{1}{2} \cdot (1) \cdot \ln(1) - \frac{1}{4} \cdot (1)] - [\frac{1}{2} \cdot (0) \cdot \ln(0) - \frac{1}{4} \cdot (0)] = -\frac{1}{4}\)

Step 5: Verification

Check the obtained integral value using a graphing utility if it yields the same result. This step is essential to confirm the accuracy of your integrated value. In this case, the integral of \(x \ln x\) from \(0\) to \(1\) is indeed equal to \(-\frac{1}{4}\)