## anonymous 5 years ago If f prime (x) = sin(pi e^x/2) and f(0) = 1 then f(2) = ?

1. anonymous

$f \prime (x) = \sin(\pi e^x/2) and f(0)=1 then f(2)=?$

2. anonymous

This is just like the other one. Knowing what integrals do.

3. anonymous

Integrals take the antiderivative of a function. Since f' is the first derivative of a function, taking the antiderivative of it gives the original function.

4. anonymous

Well considering theres a sin, pi, and e^x in there, its a bit intimidating! I'm not sure what to do with the pi or e

5. anonymous

Nothing, they're just constants. Since you're not actually having to calculate the integrals by hand, there's nothing more complicated about this than if you were given f'(x) = x Just remember that after you integrate, you add the constant C. That's needed to find the original f(x). Once you integrate, you have the equation f(0) = 1. Use it to make your f(x) by pluging in 1 for the resulting f(x) and 0 for your x value(s). Solving for C will let you have the full f(x) equation. After that, you just plug in 2 for your x values in f(2) to get your answer.

6. anonymous

Don't let the form of an integral intimidate you, it's all the same stuff.

7. anonymous

So would it just be $\cos (\pi e^x/2)$ or do I need to do something else?

8. anonymous

9. anonymous

Take the integral of it, not the derivative. It should return something different, as the derivative of cos(pi*e^x/2) is not sin(pi*e^x/2)

10. anonymous

Well then I'm confused.

11. anonymous

$\int\limits_{?}^{?}sin(pi*e^x/2 dx$

12. anonymous

I'm still lost

13. anonymous

I thought the integral would be $- \cos(\pi e^x/2) + c$ but now I'm confused...

14. anonymous

Do you know chain rule?

15. anonymous

I've learned it before but I don't really remember it

16. anonymous

In this situation it's applied like so: For a trig function, you take the derivative of it as a whole(which is what you did) and multiply it by the derivative of the inside. If you took the derivative of your answer using chain rule, it wouldn't be the original equation, so it can't be correct.

17. anonymous

Try using u substitution on the integral before solving to simplify things.

18. anonymous

u substitution just makes things more complicated... Isn't there a quick, easy way to solve this?

19. anonymous

Not that I know of. It's actually not hard when you pick the right u. With the right u, it will end up in this form: http://en.wikipedia.org/wiki/Sine_integral#Sine_integral Then you just resubstitue and you're done with the hardest part. Then it's just a matter of finding f(2)