Question

If $F\left(x\right)={\int }_{x^2}^{x^3}\log tdt(x>0)$ , then $F^1\left(x\right)=$

• A. $(9x^2-4x)\log x$
• B. $(4x-9x^2)\log x$
• C. $(9x^2+4x)\log x$
• D. $(4x^2-9x)\log x$

Answer 1

The correct option is A $(9x^2-4x)\log x$

$F\left(x\right)={\int }_{x^2}^{x^3}logtdt(x>0)$ then
$F^{{}^{\prime }}\left(x\right)$
$F^{{}^{\prime }}\left(x\right)=[log{x}^3\cdot \frac{d{x}^3}{dx}-log{x}^2\frac{d{x}^2}{dx}]$
$=\left[3logx\right(3x^2)-2logx2x]$
$=9\left(logx\right)x^2-hlogx\left(x\right)$
$=(9x^2-4x)logx$