How do you evaluate the integral cscθ?

1 Answer
Mar 15, 2017

csc(θ)dθ=ln|csc(θ)+cot(θ)|+C

Explanation:

If you know the process of finding sec(θ)dθ, this will be very similar.

Very unintuitively, make the modification:

csc(θ)dθ=csc(θ)(csc(θ)+cot(θ)csc(θ)+cot(θ))dθ=csc2(θ)+csc(θ)cot(θ)csc(θ)+cot(θ)dθ

Now let u=csc(θ)+cot(θ). Knowing their derivatives, we can say that du=(csc(θ)cot(θ)csc2(θ))dθ.

Note that the numerator of the integral is just the derivative of its denominator times 1.

=csc(θ)cot(θ)csc2(θ)csc(θ)+cot(θ)dθ=duu=ln|u|

Reversing the substitution:

csc(θ)dθ=ln|csc(θ)+cot(θ)|+C