Question #5240a

1 Answer
Mar 27, 2017

#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))#