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 May 2nd, 2017, 06:56 PM #1 Member   Joined: Feb 2017 From: henderson Posts: 36 Thanks: 0 Use the Fundamental Theorem of Calculus to calculate the definite integral. Give an e So I am having a problem with using the fundamental theorem for this problem but I might have messed up with the substitution part I know I took the derivative and I know its wrong so I tried the anti way
 May 2nd, 2017, 07:20 PM #2 Senior Member   Joined: Aug 2012 Posts: 2,099 Thanks: 604 I didn't follow your work but at the bottom you have $e^{\sin x} \cdot \sin x$. That can't be right. If you differentiate $e^{\cos x}$ you'll get $e^{\cos x} \cdot \sin x$ give or take some minus signs I was too lazy to account for. But once you do that you'll have the correct antiderivative. Ok let me be unlazy. $\frac{d}{dx} e^{-\cos x} = e^{-\cos x} \cdot (-1)(-\sin x) = e^{-\cos x} \cdot \sin x$. This is exactly your integrand so the definite integral is $e^{-\cos x} \biggr \rvert_0^{\frac{\pi}{2}} = e^0 - e^1 = 1 -e$ if I did that right. I should mention that I didn't use a formal substitution, I just eyeballed the integrand and saw that it was set up to be easy. Last edited by Maschke; May 2nd, 2017 at 07:40 PM.
 May 2nd, 2017, 07:33 PM #3 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond $$\int e^{-\cos\theta}\sin\theta\,d\theta=\int e^u\frac{du}{d\theta}d\theta=\int e^u\,du=e^u+C=e^{-\cos\theta}+C$$
May 2nd, 2017, 07:57 PM   #4
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Quote:
 Originally Posted by greg1313 $$\int e^{-\cos\theta}\sin\theta\,d\theta=\int e^u\frac{du}{d\theta}d\theta=\int e^u\,du=e^u+C=e^{-\cos\theta}+C$$
how do you use the Fundamental Theorem of Calculus on the problem?

 May 2nd, 2017, 08:07 PM #5 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,885 Thanks: 1088 Math Focus: Elementary mathematics and beyond FTOC: $$\int_a^bf(x)\,dx=F(b)-F(a)$$ where $F(x)$ is the antiderivative of $f(x)$. Can you work from there? Last edited by greg1313; May 2nd, 2017 at 10:16 PM.

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