Elizabethh58413: Created page with " $\text{use Exercise 47 to evaluate} \int(\ln{x})^3dx$

$\text{Exercise 47:}\qquad x(\ln{x})^n-n\int(\ln{x})^{n-1}dx$

$\int(\ln{x})^3dx= x(\ln{x})^3-3\int(\ln{x})^2dx$

$u = (\ln{x})^2 \qquad dv = dx$

$du = 2\frac{1}{x}\ln{x}dx \qquad v = x$

$\begin{align} \int(\ln{x})^3dx &= x(\ln{x})^3-3\int(\ln{x})^2dx = x(\ln{x})^3-3[x(\ln{x})^2 - 2\int(\ln{x})dx] \\[2e..."$

2022-11-29T03:55:56Z

Created page with "<math> \text{use Exercise 47 to evaluate} \int(\ln{x})^3dx </math> <math> \text{Exercise 47:}\qquad x(\ln{x})^n-n\int(\ln{x})^{n-1}dx </math> <math> \int(\ln{x})^3dx= x(\ln{x})^3-3\int(\ln{x})^2dx </math> <math> u = (\ln{x})^2 \qquad dv = dx </math> <math> du = 2\frac{1}{x}\ln{x}dx \qquad v = x </math> <math> \begin{align} \int(\ln{x})^3dx &= x(\ln{x})^3-3\int(\ln{x})^2dx = x(\ln{x})^3-3[x(\ln{x})^2 - 2\int(\ln{x})dx] \\[2e..."

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<math> \text{use Exercise 47 to evaluate} \int(\ln{x})^3dx </math> 
<math> \text{Exercise 47:}\qquad x(\ln{x})^n-n\int(\ln{x})^{n-1}dx </math> 
<math>
\int(\ln{x})^3dx= x(\ln{x})^3-3\int(\ln{x})^2dx </math> 
<math> u = (\ln{x})^2 \qquad dv = dx </math> 
<math> du = 2\frac{1}{x}\ln{x}dx \qquad v = x </math> 

<math>
\begin{align}
\int(\ln{x})^3dx &= x(\ln{x})^3-3\int(\ln{x})^2dx = x(\ln{x})^3-3[x(\ln{x})^2 - 2\int(\ln{x})dx] \\[2ex]
&= x(\ln{x})^3-3[x(\ln{x})^2 -2(x\ln{x} - \int1dx) = x(\ln{x})^3-3[x(\ln{x})^2 -2x\ln{x} + 2x] \\[2ex]
& =x(\ln{x})^3 - 3x(\ln{x})^2+ 6x\ln{x} - 6x +c \\[2ex]
\end{align}
</math>

7.1 Integration By Parts/51 - Revision history