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Solving several problems in higher mathematics for freshmen
1, if the numerator of the integrand is changed to E X+ 1-E X, then the integrand is equal to 1-E X/( 1+E X), so the original formula is = ∫ DX-E X/(65433).

=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.