I was given the following problem by my instructor:

“Let g (x) be a specific antiderivative of $f(x)$, that is,

$g′ (x) = f (x)$

Then we can write

$\int\_a^xf(x)dt=g(x)$ for some $a$

(This is in your book, so don’t go denying it!)  

So look at this. Let $g (x) = \sin x + 5$, and $f (x) = \cos x$. Clearly $g′ (x) = f (x)$. The Fundamental
Theorem of Calculus then says that we can write

$\int\_a^x \cos t dt = g (x)$  

$\sin x − \sin a = \sin x + 5$  

$\sin a = −5$

But $\sin a$ is never $-5$!

Find the error.

I believe they got it from [this site](http://uni.dougshaw.com/findtheerror/FTEanti.html). Can someone give me some help?