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Prove that
Solution:
Consider the given equation above
Rewrite the left side of the equation as a reciprocal of trigonometric function as follows
Since the above equation cannot be integrated by a simple integration, then we have to use the principles of trigonometric identities as follows
Multiply both the numerator and the denominator by sin x, we have
But
Hence, the above equation becomes
If
then
Integrate the above equation using algebraic substitution as follows
Express the right side of the equation into partial fractions, we have
Solve for the value of A and B by equating their coefficients:
for u:
for constant:
Substitute the first equation to the second equation
and
Substitute the value of A and B to the original equation, we have
But
Hence, the above equation becomes
Multiply both the numerator and the denominator by 1 - cos x, we have
But
and the above equation becomes
Therefore,
