Question

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

Short 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-by-step solution

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 \(v = (x^3/3 + 4x)\).

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.

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 (\