1,  substituting the complex part in the function:

f(x)=x/(1+x^2), let u=1+x^2 , du = 2x dx . There is another strategy that we can tell whether a function is suitable for substitution by thinking what the derivative of the substituted part will look like.  In the case x/(1+x^2), we should notice that the derivative of x^n will lead to nx^(n-1) which implies that the numerator should also contain some similar structure in order to simplify the function. With the same idea, when we substitute sin(x) (cos (x)) we should expect the other part of the function to have cos(x) (sin (x))

2. substitution of trigonometric function:

for functions like f(x)=(sin(x))^m (cos(x))^n or (tan(x))^m (sec(x))^n , we can always do a substitution using the characteristics of their derivative (sin(x))’=cos(x)  (cos(x))’= – sin(x) (tan(x))’ = 1/sec^2(x)  (sec(x))’=sec(x)tan(x).

To be more specific, we divide these functions into two situations:

a) If m and n are all even, then we  can always use 1 + tan^2(x)=sec^2 (x) , sin^2 (x) + cos^2 (x)= 1, cos^2 (x) – sin^2 (x)= cos(2x) to make functions be simplified into the one with single name of trigonometric function and substituted.

b) If one of m or n is odd, then for the trig function with odd power, we “drag out” one of it to make up a form like tan^3 (x)=tan^2 (x) * tan(x), and then the part which is to the power of even number can again be substitute by using  1 + tan^2(x)=sec^2 (x) , sin^2 (x) + cos^2 (x)= 1 and so the function is again simplified into the one with single name of trigonometric function.

3. substitution using trig function

For the function with a part (x^2 + a^2)  (x^2 – a^2) (a^2 + x^2) (a^2 – x^2), we can try to substitute x with asin(x) acos(x)  atan(x) , and therefore simplify the function using  1 + tan^2(x)=sec^2 (x) , sin^2 (x) + cos^2 (x)= 1