Answered: Find ∫xn ln x dx for n = 1, 2, and 3.…

Name: Find ∫xn ln x dx for n = 1, 2, and 3. Describe any patterns you notice
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Description: Find ∫xn ln x dx for n = 1, 2, and 3. Describe any patterns you notice.

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter6: Exponential And Logarithmic Functions

Section6.3: Logarithmic Functions

Problem 61SE: Is there a number x such that ln x=2 ? If so, whatis that number? Verify the result.

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Question

Find ∫xn ln x dx for n = 1, 2, and 3. Describe any patterns you notice.

Expert Solution

Step 1

Consider the following integral:

$\int x^{n} \ln (x) d x . . . . . . (1)$

Substitute $n = 1$ in $(1)$ :

$\int x \ln (x) d x$

Use integration by parts:

$\begin{array}{rcl} \int f g' & = & f g - \int f' g \\ f & = & \ln (x), g' = x \\ f' & = & \frac{1}{x}, g = \frac{x^{2}}{2} \end{array}$

Step 2

$\begin{array}{rcl} \int x \ln (x) d x & = & \frac{x^{2} \ln (x)}{2} - \int \frac{x}{2} d x \\ = & \frac{x^{2} \ln (x)}{2} - \frac{x^{2}}{4} + C \\ = & \frac{2 x^{2} \ln (x) - x^{2}}{4} + C \\ = & \frac{x^{2} [2 \ln (x) - 1]}{4} + C \end{array}$

Substitute $n = 2$ in $(1)$ :

$\int x^{2} \ln (x) d x$

Use integration by parts:

$\begin{array}{rcl} \int f g' & = & f g - \int f' g \\ f & = & \ln (x), g' = x^{2} \\ f' & = & \frac{1}{x}, g = \frac{x^{3}}{3} \end{array}$

Step 3

$\begin{array}{rcl} \int x^{2} \ln (x) d x & = & \frac{x^{3} \ln (x)}{3} - \int \frac{x^{2}}{3} d x \\ = & \frac{x^{3} \ln (x)}{3} - \frac{x^{3}}{9} + C \\ = & \frac{3 x^{3} \ln (x) - x^{3}}{9} + C \\ = & \frac{x^{3} [3 \ln (x) - 1]}{9} + C \end{array}$

Substitute $n = 3$ in $(1)$ :

$\int x^{3} \ln (x) d x$

Use integration by parts:

$\begin{array}{rcl} \int f g' & = & f g - \int f' g \\ f & = & \ln (x), g' = x^{3} \\ f' & = & \frac{1}{x}, g = \frac{x^{4}}{4} \end{array}$

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