Question

Evaluate the definite integral. $$ \int_{1}^{4} x \ln x d x $$

Short answer

The definite integral of \(x \ln x\) from 1 to 4 is \(4 \ln(4) + 3.75\).

Step-by-step solution

Identify \(u\) and \(dv\)

Let \(u\) be \(ln(x)\) and \( dv\) be \(x dx\). These parts are selected this way to make the integral simpler, as the derivative of \(ln(x)\) is easier to manage.

Calculate \(du\) and \(v\)

Now compute \(du\) as the derivative of \(u\), and \(v\) as the antiderivative of \(dv\). We obtain that \(du = \dfrac{1} {x} dx\) and \(v = \dfrac{x^2} {2}\).

Apply Integration by Parts

Substitute \(u\), \(v\), \(dv\) and \(du\) into the integration by parts formula \(\int u dv = u \cdot v - \int v du\). This simplifies to \(\int_{1}^{4} x \ln x dx = [\ln(x) \cdot \dfrac{x^2} {2}]_{1}^{4} - \int_{1}^{4} \dfrac{x^2} {2} \cdot \dfrac{1} {x} dx\). Simplifying the second integral, obtain \(\int_{1}^{4} \dfrac{x} {2}dx\).

Simplify Integral and Evaluate

Perform integration of the simpler integral, and substitute the limits of integration. The result is: \(\int_{1}^{4} \dfrac{x} {2} dx = [\dfrac{x^2} {4}]_{1}^{4} = 4-0.25 = 3.75\). After that, substitute limits of integration into the part of the integration by parts formula: \([\ln(x) \cdot \dfrac{x^2} {2}]_{1}^{4} = [4 \ln(4) -1 \ln(1)] = 4 \ln(4)\). The final result will be the sum of these two components.

Obtain Final Result

The final step is to add the results obtained in the previous step. So, \(\int_{1}^{4} x \ln x dx\) = \(4 \ln(4) + 3.75\).