Fundamental Theorem of Calculus Questions

akbarali

New member
Last one for the night!

These are the questions:

This is my work:

I think question 5 is correct (I hope), but I'm not entirely sure about question 6. Any help would be appreciated!

MarkFL

Staff member
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?

akbarali

New member
Hmm, are you saying the answer should just be exp(sin(x)+ x^x) ?

MarkFL

Staff member
Yes, your formula doesn't have $h'(x)$ in it, right?

akbarali

New member
Is there any way you would work these problems differently from how I did that would help me better solve such problems in the future?

MarkFL

Staff member
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:

$$\displaystyle \frac{d}{dx}\left(\int_0^x e^{\sin(s)+s^s}\,ds \right)=e^{\sin(x)+x^x}$$

akbarali

New member
You have been helping me all night. So grateful for your work. I have two more problems that are bugging the heck out of me, but I think I should let you rest for the night! Hehe