1. ## Basic questions about integration by substitution

1. How does it possible for sin(x)cos(x) dx to have two different solutions, namely (1/2)*(sin2x)+c and -(1/2)*(cos2x)+c, so that would bring to (rediculous) conclusion that sin2x+cos2x=0?
2. Integration by substitution makes use of differentials (not simply derivatives), thus is it purely anti-derivative?

2. ## Re: Basic questions about integration by substitution

Originally Posted by jojo7777777
1. How does it possible for sin(x)cos(x) dx to have two different solutions, namely (1/2)*(sin2x)+c and -(1/2)*(cos2x)+c, so that would bring to (rediculous) conclusion that sin2x+cos2x=0?
You are assuming that the 2 "c"s are the same. Yes, $\int sin(x)cos(x) dx= (1/2)sin^2(x)+ c$, for some constant, c. Yes, $\int sin(x)cos(x)dx= -(1/2)cos^2(x)+ C$ where "c" and "C" are generally NOT the same. Now that reduces to $sin^2(x)+ cos^2(x)=c- C$ which tells us that $sin^2(x)+ cos^2(x)$ is a constant- NOT "ridiculous".

2. Integration by substitution makes use of differentials (not simply derivatives), thus is it purely anti-derivative?