I'm mostly curious as to if the way I've went about solving this is correct, or if there is a more simple way to get the answer.

So I first evaluated the top section

Step 1

And when I did that I got sin(e^x)

With that, I replaced the upper bound of the integral with sin(e^x) and integrated sin(t^2) dt

My Resulting equation was

Step 2

Which I then tried to differentiate using the product rule of three parts first being (2sin(e^x)), second sec(sin^2(e^x)), and the third tan(sin^2(e^x)). The resulting derivative I got was

enter image description here

So thats what i've done, any tips or suggestions would be appreciated mostly was hoping to get some feedback about the method I used and if its the best/easiest method for these types of problems.

  • $\begingroup$ You actually can't integrate $\sec(t^2)$ (the anti-derivative doesn't have a nice form), so the FTOC is definitely required $\endgroup$ – Dylan Dec 2 '14 at 2:34

Use the Fundamental Theorem of Calculus.

$$\dfrac{d}{dx} \int_a^{f(x)} G'(t) dt = \dfrac{d}{dx} \left(G(f(x))-G(a)\right) = G'(f(x))\cdot f'(x)$$

Here, $G'(t) = \sec(t^2)$ and $f(x)=\sin(e^x)$


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