# Question 5240a

Mar 27, 2017

$\csc \left(x \ln \left(x\right)\right) = - \csc \left(x \ln \left(x\right)\right) \cot \left(x \ln \left(x\right)\right) \cdot \left(\ln \left(x\right) + 1\right)$

#### Explanation:

$\csc \left(u \left(x\right)\right) ' = - \csc \left(u \left(x\right)\right) \cot \left(u \left(x\right)\right)$
$\ln \left(x\right) ' = \frac{1}{x}$
Formula for multiplication: $\left(f \cdot g\right) \left(a\right) = f ' \left(a\right) g \left(a\right) + g ' \left(a\right) f \left(a\right)$
$\left(x \ln \left(x\right)\right) ' = \frac{d}{\mathrm{dx}} \left(x\right) \cdot \ln \left(x\right) + \frac{d}{\mathrm{dx}} \ln \left(x\right) \cdot x = \left(\ln \left(x\right) + 1\right)$

$\csc \left(x \ln \left(x\right)\right) = - \csc \left(x \ln \left(x\right)\right) \cot \left(x \ln \left(x\right)\right) \cdot \frac{d}{\mathrm{dx}} \left(x \ln \left(x\right)\right)$
$\csc \left(x \ln \left(x\right)\right) = - \csc \left(x \ln \left(x\right)\right) \cot \left(x \ln \left(x\right)\right) \cdot \left(\ln \left(x\right) + 1\right)$

$- - - - - - - - - - - - - - - - - - - - -$
to prove it: $\int$ $\left(- \csc \left(x \ln \left(x\right)\right) \cot \left(x \ln \left(x\right)\right) \cdot \left(\ln \left(x\right) + 1\right)\right) \mathrm{dx}$
$u = x \ln x$
$\mathrm{du} = \left(\ln \left(x\right) + 1\right)$
$\int$ (-csc(u)cot(u ) = csc(u)#

$\int$$\left(- \csc \left(u\right) \cot \left(u\right) \cdot \mathrm{du}\right) = \left(\csc \left(u\right)\right)$
= $\csc \left(x \ln \left(x\right)\right)$