Evaluate the integral.
$ \displaystyle \int \csc x dx $
$-\ln |\csc x+\cot x|+C$
this problem is from Chapter seven section to Problem number thirty nine in the book Calculus Early Transcendental Sze. In addition by James Store, here we have a indefinite integral of Kosi Can FX. So one way to proceed is to multiply this coast, seeking of eggs and divided by the same number. So we're not really changing the immigrant, So let's multiply and divide by Kosi Can IX plus Co Tension X Again? We haven't changed the inner rule because all we have done is multiplied co sickened by one. So here for the next step, you could multiply out that numerator and we get Kosi can squared of X plus Kosi can of X Times co attention of X and the denominator remains the same. Okay, for our next step, let's apply u substitution. Let's take you to be the denominator costigan of X plus contention of X so that do you becomes negative. Kosi can of X time's cozy in genetics minus Kosi can square of X, the ex equivalently negative Do you is our room writer because he can contention. So if we apply this u substitution, we have negative in Rule one over. You do you so we can use the power rule. Here we get a negative natural log of the absolute value of you, plus our constant of integration. See? And finally we convey back substitute you using our e substitution to have natural negative natural algorithm of absolute value. Of course he can of eggs, plus coach, engine of X plus he And that's our answer.