At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
\[f \prime (x) = \sin(\pi e^x/2) and f(0)=1 then f(2)=?\]
This is just like the other one. Knowing what integrals do.
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.
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
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.
Don't let the form of an integral intimidate you, it's all the same stuff.
So would it just be \[\cos (\pi e^x/2)\] or do I need to do something else?
forgot the "+ c" already
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)
Well then I'm confused.
I'm still lost
I thought the integral would be \[- \cos(\pi e^x/2) + c\] but now I'm confused...
Do you know chain rule?
I've learned it before but I don't really remember it
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.
Try using u substitution on the integral before solving to simplify things.
u substitution just makes things more complicated... Isn't there a quick, easy way to solve this?
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)