Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

integrate

$$ \int \sin(x) \cos(x)\; dx $$

using $u$-substitution.

If i take $u = \sin(x)$ I get final answer to be $\sin^2(x) / 2 + c$

But If i take $u = \cos(x)$ I get final answer to be $-\cos^2(x) / 2 + c$

Are they equal? They should be, otherwise it does not make sense. But how are they equal?

share|improve this question
7  
Do they differ by a constant? –  David Mitra Dec 16 '12 at 3:08
    
Note that the $c$ in the first answer need not be the same as the $c$ in the second answer. –  asmeurer Dec 16 '12 at 6:39

2 Answers 2

up vote 6 down vote accepted

The symbol $\int f(x) dx$ does not denote a function, but rather the set of all functions $F(x)$ that satisfy $F'(x)=f(x)$. Hence, any two functions in this set may differ by a constant. In your particular example, note that $\sin^2(x)/2 -(-\cos^2(x)/2)=1/2$.

share|improve this answer

Remember that for two derivable functions $\,f(x)\,,\,g(x)\,$ on some open interval $\,I\,$, we have that

$$\forall\,x\in I\;\;,\;f'(x)=g'(x)\Longleftrightarrow f(x)=g(x)+C\,\,,\,C=\,\text{ a constant}$$

Now apply the above to your problem...

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.