#Integration help
cosmic sinew
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thorny quest
Try substitution with x = tan(t)
thorny quest
When there are quotients with trigonometric functions in the denominator,
it is hard to integrate
cosmic sinew
Wait but 1+x^2 isnt a trigonometric function 😭
thorny quest
so we try to simplify them in order to make the functions appear solely in the numerator
thorny quest
Wait but 1+x^2 isnt a trigonometric function 😭
Wait you are right 
I stand corrected
"When there are rational functions in the integrand with a multinomial (not monomial) in the denominator,"
Then the 1+x^2 motivates us to pick tan(t), since 1 + tan^2(t) = sec^2(t)
cosmic sinew
Im sorry I dont really understand how or why you can plug in tan(x) could you please explain it
thorny quest
We are not exactly "plugging in", we are subtituting in x = tan(t) (note: note tan(x) either)
We have to transform the domain of integration as well
We have originally $$\int_{-\infty}^\infty\frac{1}{1+x^2}\text{ d}x$$
formal elkBOT
ℝafain
thorny quest
Apart from the integrand, we have to transform the "differential" dx into x'(t) dt and the bounds for integration
cosmic sinew
Okay I think I understand but why are we plugging in x = tan(t)
cosmic sinew
Yeah I understand that but what is the reasoning behind that, I dont understand what principle we use to do that
thorny quest
It's not exactly a rule, just educated guesses generated from experience
thorny quest
so we try to simplify them in order to make the functions appear solely in the n...
The motivation is here
If it doesn't work, we adapt and improvise again
cosmic sinew
Ohh so trial and error?
thorny quest
Well people have tried ahead of us, so we need not err anymore
cosmic sinew
Im sorry I sould like a broken record here but what about the given value motivated you to try tan first?
cosmic sinew
Yes but what about the 1+x^2
thorny quest
If we substitute x = tan(t) into 1 + x^2, then that becomes sec^2(t), which can be divided by other trig functions to retain the form sin^m(x) cos^n(x) for some integers m and n
unlike rational functions with denominator 1 + x^2
That "rational function" form of trig functions is considerably easier to integrate
thorny quest
It's just easier to integrate "monomial" over "monomial" forms of rational functions
cosmic sinew
and then turn that into cos^2(t) and take it from there?
thorny quest
As you can see, the powers get to be cancelled out
thorny quest
and then turn that into cos^2(t) and take it from there?
No, try transforming dx into x'(t) dt
There are cancellations to be done