Can i split it up and do 1/(1+x^2) dx and arctanx dx? then multiply the results?
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And the quotient rule is just a special case of the product rule. You might integrate the square of cosec x by working back through it. Balloon Calculus: standard integrals and derivatives
While I'm here, here's (or will be in two tics) a picture for the present (chain rule) problem, just in case it helps...
Don't integrate - balloontegrate
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I agree that int by parts is a product rule of sorts for certain types of products there is no product rule that works on every product as with differentiation.
In the same light partial fraction decompostion is a quotient rule if you will for rational functions but again there is no integration rule for quotients in general as with differentiation
No No No
Lets look at this again
when you take u = arctan(x)
du = 1/(1+x^2)dx which yields (1+x^2)du = dx
int(arctan(x)/(1+x^2)dx = int (u (1+x^2)/(1+x^2)dx) = int(udu) which can be easily integrated
If you let u = 1 + x^2 then du = 2x dx
you get int (arctan(x)/(2xu)du) which is a mess.
The idea of u substitution is to obtain an integral in terms of u which you can integrate.
In general there is only one choice maybe two which will accomplish this.
What substitution to make ? thatg comes from experience , insight, and good ol' trial and error.
Ok the method is called substitution because we substitute
we have int (arcatan(x)/(1+x^2)dx
We'll go one step at a time
Since u = arctan(x) substitute arctan(x) with u
int (arcatan(x)/(1+x^2)dx = int(u/(1+x^2)dx)
Now dx = (1+x^2)du so replace dx with this result
int(u/(1+x^2)dx) = int(u/(1+x^2) *(1+x^2)du)
Note (1+x^2)/(1+x^2) cancels leaving int(udu)