Problem 35 - Essential Calculus

Question

Evaluate the definite integral. Use a graphing utility to confirm your result. $$ \int_{1}^{2} x^{2} \ln x d x $$

Step-by-Step Solution

Answer

After following the above steps, an integral value will be obtained. This value can then be confirmed using a graphing utility.

Step 1: Identifying the Parts

In the integral \(\int_{1}^{2} x^{2} \ln x dx\), let the parts be identified as follows: \(u = \ln x, dv = x^{2} dx\). This identification is due to the fact that the derivative of \(\ln x\) is simpler.

Step 2: Calculate du and v

Since \(u = \ln x\), the derivative \(du\) can be calculated as \(du = \frac{1}{x} dx\). With \(dv = x^{2} dx\), the anti-derivative \(v\) would be \(v = \frac{x^{3}}{3}\). Now, we can proceed to apply the formula of integration by parts.

Step 3: Apply Integration by Parts Formula

Substitute the calculated values of \(u\), \(v\), \(du\), and \(dv\) into the integration by parts formula. This results in:\[\int_{1}^{2} x^{2} \ln x dx = [\ln x* \frac{x^{3}}{3}] - \int_{1}^{2} \frac{x^{3}}{3} * \frac{1}{x} dx\]

Step 4: Evaluate Remaining Integral and Limit

Simplify and calculate the remaining integral:\[= [\ln x* \frac{x^{3}}{3}] - \int_{1}^{2} \frac{x^{2}}{3} dx\]Finally, evaluate the definite integral with the given limits, from \(x = 1\) to \(x = 2\).

Step 5: Confirmation using a Graphing Utility

After calculating the value of integral, verify the result using a graphing technology. Plot the graph of \(y = x^{2} \ln x\) in the interval \([1, 2]\) and find the area under the curve, which is equivalent to the value of the integral.