Sometimes in order to find integral I was taught to substitute the variable with a suitable trigonometric function. For example if the denominator is in the form $(a^2+x^2)$ we tend to replace $x$ with $\tan\theta$ and then proceed with integration. Again for some cases we replace the variable with $\sin\theta/\cos\theta$. But isn't that wrong to do so? Because the value of variable can be anything, it could be more than $1$ as well as less than $-1$. So when we are replacing that variable with $\sin\theta$ shouldn't it be wrong Since value of $\sin\theta$ cannot be more than $1$? Again if $\tan\theta$ is used in some cases we might obtain the integral in the form of an inverse circular function. But while considering the replacement we are limiting the integral(in the form of inverse trig functions) within the principle value. So is this correct too?
Lets say we encountered a problem which requires integral of a certain function. One can find the integral in numerous ways. However the format of the equation is such that if we replace the variable with a trigonometric function, it becomes a lot easier and shorter to find the integral. My question is that if we are to replace the variable with a suitable trigonometric function, there might be for some value of the variable, value of trigonometric function might not exist. So is this a correct way to solve such problems?
I know for most of the cases this way(above) is correct kust being curious if there might be any exception or anything.
Now the second question is that if we solve in that way and the integral turns out to be a inverse circular function like $a\sin^-1\theta +c$ then aren't we limiting the integral within the bounds of principle values of theta?
I am asking this question just out of curiousity. I might have missed some crucial things which wouldn't have led me to these questions