Problem 62 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to evaluate the integral. $$ \int_{1}^{4} \ln x\left(x^{2}+4\right) d x $$

Step-by-Step Solution

Answer

The integral \(\int_{1}^{4} \ln x\left(x^{2}+4\right) dx\) equals \([(\ln x)(\frac{x^3}{3} + 4x)]_1^4 - [\frac{x^3}{9} + 2x^2]_1^4\), which after evaluation gives a numerical answer.

Step 1: Express the Integral in Suitable Form for Integration by Parts

The integral \(\int_{1}^{4} \ln x\left(x^{2}+4\right) dx\) is in a form suitable for integration by parts. By applying the integration by parts formula \(\int u dv = uv - \int v du\), let's define \(u = \ln x\) and \(dv = (x^2 + 4) dx\). Then, the derivatives and integrals needed for the formula are: \(du = (1/x) dx\) and \(v = (x^3/3 + 4x)\).

Step 2: Apply the Integration by Parts Formula

By substituting \(u, du, v, dv\) into the integration by parts formula, the integral becomes \(\int_{1}^{4} \ln x\left(x^{2}+4\right) dx = [(\ln x)(\frac{x^3}{3} + 4x)]_1^4 - \int_{1}^{4} (\frac{x^3}{3} + 4x)\cdot (\frac{1}{x}) dx\), which simplifies to \([(\ln x)(\frac{x^3}{3} + 4x)]_1^4 - \int_{1}^{4} (\frac{x^2}{3} + 4) dx\). We can integrate the simple polynomial in the second term directly.

Step 3: Evaluate the Definite Integral

Now, we calculate the remaining integral in the second term, and evaluate the result at the limits of integration (1 and 4). Specifically, we have \([(\ln x)(\frac{x^3}{3} + 4x)]_1^4 - [\frac{x^3}{9} + 2x^2]_1^4\). After the calculation we obtain a numerical answer.

Key Concepts

Integration by Parts

Integration by parts is a potent technique used for finding integrals where the standard methods are not applicable. It's particularly useful when dealing with integrals involving products of functions, such as \(\int u dv\). To apply it correctly, choose two parts of the integrand: one to differentiate (\