Discussion of
csc x = - ln|csc x + cot x| + C.
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1. Proof
Strategy: The strategy is not obvious. Multiply and divide
by (csc x + cot x); use Substitution.
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csc x dx =
csc x |
csc x + cot x
csc x + cot x |
dx |
set
u = csc x + cot x
then we find
du = (- csc x cot x - csc2 x) dx
substitute du = (- csc x cot x - csc2 x) dx, u = csc x +
cot x
|
csc x |
csc x + cot x
csc x + cot x |
dx = - |
(- csc2 x - csc x cot x) dx
csc x + cot x
|
| = - |
du
u |
solve integral
= - ln |u| + C
substitute back u=csc x + cot x
= - ln |csc x + cot x| + C
Q.E.D.
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