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Chapter 6: Problem 12

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

Short Answer

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

Step by step solution

01

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'.
02

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 \(v = \frac{1}{5} x^{5}\).
03

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: \(ln x (\frac{1}{5} x^{5}) - \frac{1}{5} \int x^{4} dx\)
04

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: \(\frac{1}{5} \int x^{4} dx = \frac{1}{5} * \frac{1}{5} x^{5} = \frac{1}{25} x^{5}\)
05

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.

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