=x-∫d( 1+e^x)/( 1+e^x)=x-ln( 1+e^x)+c
2. Let sqrt (x+ 1) = t and x = t 2- 1, then the original formula = ∫ 2dt/(t2-1) = ∫ dt/(t-1).
Then replace t with x, that is, ln (sqrt (x+1)-1)-ln (sqrt (x+1)+1)+c (sqrt stands for root).
3. Using the first-order differential invariance of unary function, we can calculate: df (x2) = df (x2)/d (x2) * d (x2) = f' (x2) * 2xdx = 2sin (x2)/xdx.
You can also use the method of compound derivation, but it is essentially the same. The answer is as follows.