Hey guys i kind of have a weird little problem which i can't figure out.
I'm integrating sinxcosx, and depending on what i use as my substution you get two answers.
1) sin^2x/2 + C and 2) -cos^2x/2 + C
Now if we set these two answers equal to eachother we get
sin^2x + cos^2x = 0 but we know there exists a trig identity that states sin^2x + cos^2x = 1
So i'm confused here. How can this be possible? I know that 0 cannot equal 1. Or at least i hope not hehe. Any help would be great cheers
And I think the problem you have you think you get two distinct answers when you use and when you switch to .
But in reality these are the same (the set of all anti-derivates) because,
I just renamed the constant function,
Thus, you get the same thing.
And the last thing you could have done is,
I just multiplied and divide by 2, unchanged expression.
From trignometry you know that,
But again though it looks different the space of solutions is still the same. You can use some trignometry to confirm this.
This is a classic puzzler.
I ran across it across it in Calculus I.
There are three ways (at least) to integrate it.
We have: .
 If we look at it as: .
. . . we get: .
 If we write it as: .
. . . we let:
Since these two answers must be equal, we have:
Multiply by 2: .
The time the constants aren't completely arbitrary.
. . They must satisfy: .
 We can use the identity: .
. . I'll let you justify this one . . .
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A similar problem: .