Originally Posted by EarlyTravel
f(x) = 4x/(1+x^2) = 4x(1+x^2)^-1
We're now searching for the primitive function F(x), this is found to be
F(x) = 2*ln(1+x^2)+C
You can try to take the derivative of F(x) and see that it holds. When you write a function and sees ^-1, be aware of the ln(x) function since y = ln(x) => y' = 1/x = x^-1.
The same holds for 2).
f(x) = cos(x)/(1+2sin(x)) = cos(x)(1+2sin(x))^-1
You can now find
F(x) = ln(1+2sin(x))/2+C
Another tip is to start with the ln function, and then se what the derivative is by using the chainrule, then you for example know that in this case you should divide by 2 so it cancel when taking the derivative of the function F(x). But be aware! This only holds when we're talking about constants, since if we would use this strategy on a function, we would have to apply the product rule when calculating the derivative and that would almost surely lead to wrong function.