Integral of ∫csc(x)dx

Like the function sec(x), there is no obvious way to integrate csc(x). We need to manipulate csc(x) into an integratable form.

One way is to use the method of partial fractions. While this method takes a quite a few steps to solve the integral, it is robust.

Using the method of partial fractions, the idea is to rewrite the integral of csc(x) as:

∫csc(x)dx = ∫sin(x)dx / (1 - cos^2(x))

Then by letting u = cos(x)...

∫csc(x)dx = -∫du / (1 - u^2)

We can then convert the expression 1/(1+u)(1-u) into the sum of its partial fractions.

Another way involves some trickery where we try to form an integral of the form f'(x)/f(x), which gives the result ln|f(x)|. Although this method is efficient, it requires you to know part of the answer before you begin, which doesn't make much sense.

Thanks for watching. Please give me a "thumbs up" if you have found this video helpful.

Please ask me a maths question by commenting below and I will try to help you in future videos.

[matheus_rml]

This is the best explanation of this integral.

[햄쥐-cu]

Hello. I am Korean student. Your video became me very useful. Thank you so much for good explanation. I can solve many mathematics problems ascribe to you. Thank U.

[ericb]

Nice skills! Thanks for demonstrating two ways of thinking about the integral of csc(x). The only suggestion would be to incorporate the partial fraction decomposition, as it is a very important and crucial component of separating the integrand. Great work!

[robertherbert]

....wonderful clarity of an explanation starting from first principles using no tricks.

[caistead]

Hello Wu!! Fantastic work!! Keep em coming, any chance you could do a video on 3d vectors, perpendicular 3d vectors and intersections/angles between planes?

[meriemlak]

Amazing work !
Thank u for your great explanation that was so helpfull

[ketofitforlife]

You didn't leave the link for pfd

[philippecailleux]

You realize of course that this is ln|tan(x/2)|...