# Basic questions about integration by substitution

• Feb 3rd 2013, 10:54 AM
jojo7777777
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?
• Feb 3rd 2013, 11:21 AM
HallsofIvy
Re: Basic questions about integration by substitution
Quote:

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, $\displaystyle \int sin(x)cos(x) dx= (1/2)sin^2(x)+ c$, for some constant, c. Yes, $\displaystyle \int sin(x)cos(x)dx= -(1/2)cos^2(x)+ C$ where "c" and "C" are generally NOT the same. Now that reduces to $\displaystyle sin^2(x)+ cos^2(x)=c- C$ which tells us that $\displaystyle sin^2(x)+ cos^2(x)$ is a constant- NOT "ridiculous".

Quote:

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