Question

$\int (\log x{)}^{2}dx=$

• A. $x\left[\right(\log x{)}^2-2$ $\log x$ $+2]+c$
• B. $x\left[\right(\log x{)}^2+2$ $\log x$ $+2]+c$
• C. $\left[\right(\log x{)}^2-2$ $\log x$ $+2]+c$
• D. $\left[\right(\log x{)}^2+2$ $\log x$ $+2]+c$

Answer 1

Answer: (A)

$x\left[\right(\log x{)}^2-2$ $\log x$ $+2]+c$

$\int (\log x{)}_1^{2}dx\int (\log x{)}^2-1dx$
$(\log x{)}^2\int 1dx-\int \frac{2\left(\log x\right)}{x}xdx$
$=\left(\log x{)}^2\right[x]-2\int \log xdx$
$\int \log x=x\log x-x$
$=x(\log x{)}^2-2[x\log x-x]+c$
$=x(\log x{)}^2-2x(\log x)+2x+c$
$=x\left[\right(\log x{)}^2-2\left(\log x\right)+2]+c$
$\Rightarrow \int (\log x{)}^{2}dx=x[(\log x{)}^2-2(\log x)+2]+c$