Question

Question #5240a

Answer 1

`csc(xln(x))= -csc(xln(x))cot(xln(x) )*(ln(x) + 1)`

Explanation:

`csc(u(x))' = -csc(u(x))cot(u(x))`
`ln(x)' = 1/x`
Formula for multiplication: `(f * g)(a) = f'(a)g(a) + g'(a)f(a)`
`(xln(x))'= d/dx (x)*ln(x) + d/dx ln(x)*x = (ln(x) + 1)`

`csc(xln(x)) = -csc(xln(x))cot(xln(x) )* d/dx (xln(x))`
`csc(xln(x))= -csc(xln(x))cot(xln(x) )*(ln(x) + 1)`

`---------------------`
to prove it: `int` `(-csc(xln(x))cot(xln(x) )*(ln(x) + 1))dx`
`u = xlnx`
`du= (ln(x) + 1)`
`int` `(-csc(u)cot(u ) = csc(u)`

`int` `(-csc(u)cot(u )*du) = (csc(u))`
= `csc(xln(x))`