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Fundamental Theorem of Calculus Questions
- Thread starter akbarali
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For the first one, I agree with your result. Good work! 
For the second one, the derivative form of the FTOC gives us:
If:
\(\displaystyle G(x)=\int_a^x f(t)\,dt\)
then:
\(\displaystyle G'(x)=f(x)\)
You have cited a more general case, but can you see that you have added something to your result which should not be there?
For the second one, the derivative form of the FTOC gives us:
If:
\(\displaystyle G(x)=\int_a^x f(t)\,dt\)
then:
\(\displaystyle G'(x)=f(x)\)
You have cited a more general case, but can you see that you have added something to your result which should not be there?
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The first problem I would write:
\(\displaystyle \int_0^{\pi}\sin(x)\,dx=-\left[\cos(x) \right]_0^{\pi}=-\left(\cos(\pi)-\cos(0) \right)=-(-1-1)=-(-2)=2\)
For the second problem, I would simply write:
\(\displaystyle \frac{d}{dx}\left(\int_0^x e^{\sin(s)+s^s}\,ds \right)=e^{\sin(x)+x^x}\)
\(\displaystyle \int_0^{\pi}\sin(x)\,dx=-\left[\cos(x) \right]_0^{\pi}=-\left(\cos(\pi)-\cos(0) \right)=-(-1-1)=-(-2)=2\)
For the second problem, I would simply write:
\(\displaystyle \frac{d}{dx}\left(\int_0^x e^{\sin(s)+s^s}\,ds \right)=e^{\sin(x)+x^x}\)
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