Problem 12 - Essential Calculus

Question

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x^{4} \ln x d x $$

Step-by-Step Solution

Answer

\(\frac{1}{5} x^{5} ln x - \frac{1}{25} x^{5} + C\)

Step 1: Define functions 'u' and 'dv'

Given that the integration by parts equation is \(\int u dv = u v - \int v du\), first, we need to determine what 'u' and 'dv' are. Let \(u = ln x\) and \(dv = x^{4} dx\). This choice is made due to the simplicity when differentiating the logarithmic function to get 'du' and integrating the power function to get 'v'.

Step 2: Determine 'du' and 'v'

Now differentiate 'u' to get 'du' and integrate 'dv' to get 'v'. Differentiating 'u' \(= ln x\) , we get \(du = \frac{1}{x}dx\). Integrating 'dv' \(= x^{4} dx\), we get \(v = \frac{1}{5} x^{5}\).

Step 3: Substitute into Integration by Parts formula

Substituting 'u', 'v', 'du', and 'dv' into the integration by parts formula \(\int u dv = u v - \int v du\) , we get the integral: \(\int x^{4} ln x dx = u v - \int v du = ln x (\frac{1}{5} x^{5}) - \int (\frac{1}{5} x^{5}) (\frac{1}{x}dx)\) which simplifies to: \(ln x (\frac{1}{5} x^{5}) - \frac{1}{5} \int x^{4} dx\)

Step 4: Solve remaining integral

Now solve the remaining integral. The remaining integral \(\int x^{4} dx\) is straightforward as it is a power rule integral. Thus, the integral becomes: \(\frac{1}{5} \int x^{4} dx = \frac{1}{5} * \frac{1}{5} x^{5} = \frac{1}{25} x^{5}\)

Step 5: Form complete solution

Substitute the result back into our integrated function so the final result is: \(ln x (\frac{1}{5} x^{5}) - \frac{1}{5} (\frac{1}{25} x^{5}) = \frac{1}{5} x^{5} ln x - \frac{1}{25} x^{5} + C\), where C denotes the constant of integration.