Why can't you integrate #sqrt(1+(cosx/-sinx)^2#?

Question

William Abdalla · Accepted Answer

The answer is: #ln|tan(x/2)|+c#

First of all it's useful to "change" a little the function:

#y=sqrt(1+(cosx/-sinx)^2)=sqrt(1+(cos^2x)/(sin^2x))=sqrt((sin^2x+cos^2x)/(sin^2x))=sqrt(1/(sin^2x))=1/sinx#.

I used the first fondamental relation of trigonometry: #sin^2x+cos^2x=1#

So we have to integrate: #int1/sinxdx#.

Before integrate this function, it is useful to remember the parametric formula of sinus, that says:

#sinx=(2t)/(1+t^2)#, where #t=tan(x/2)#.

Now the integral will be done with the method of substitution:

#tan(x/2)=trArrx/2=arctantrArrx=2arctantrArrdx=2(1/(1+t^2))dt#.

Our integral becoms:

#int1/sinxdx=int1/((2t)/(1+t^2))2(1/(1+t^2))dt=int(1+t^2)/(2t)2(1/(1+t^2))dt=int1/tdt=ln|t|+c=ln|tan(x/2)|+c#

Charles Arocha · Answer

undefined The integral of $ \sqrt{1 + \left(\frac{\cos(x)}{-\sin(x)}\right)^2} $ cannot be integrated in terms of elementary functions because it does not have an antiderivative that can be expressed using standard functions like polynomials, exponentials, trigonometric functions, and their inverses. This integral involves a trigonometric expression within the square root, which leads to a complex integrand. While certain techniques like trigonometric substitution or integration by parts may be attempted, they often result in expressions that cannot be simplified further. As a result, the integral remains in a form that cannot be evaluated using standard methods, making it non-elementary.