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)

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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))