Suppose in general a pair of functions

[Flyboy27]

Suppose in general that we have two functions

$$F(x)= \int_{0}^{cos x}e^{xt^2} dt$$
$$G(x)= \int_{0}^{cos x}\(t^2e^{xt^2} dt$$
$$H(x) = G(x) - F'(x)$$

Where, I need to prove that
$$H(\frac{\pi}{4}) = e^\frac{\pi}{8}/\sqrt{2}$$

Okay, so far I have computed the integrals of both of these functions, where I am confused is when computing $$F'(x)$$ do I differentiate the integrand with respect to x only, and then simply subtract the two functions. Sorry for the edit, I left off the $$dt$$ for both integrals. Any help would be appreciated!

 

[NateTG]

Suppose in general that we have two functions

$$F(x)= \int_{0}^{cos x}e^{xt^2}$$
$$G(x)= \int_{0}^{cos x}\(t^2e^{xt^2}$$

These integrals are a bit confusing. Are they supposed to be, for example:
$$F(x)= \int_{0}^{cos x}e^{xt^2} dt$$

Or something different?

 

[Flyboy27]

Yes I corrected the original post, sorry I left off the $$dt$$ for both integrals.

 

[HallsofIvy]

To find the derivative, with respect to x, of $F(x)= \int_{0}^{cos x}e^{xt^2}dt$, use "LaGrange's Formula" [tex]\frac{d\int_{a(x)}^{b(x)} f(x,t)dt}{dx}= \int_{a(x)}^{b(x)} \frac{\partial f(x,t)}{\partial x} dt+ F(b(x))\frac{db}{dx}- F(a(x)\frac{da}{dx}[/itex].

 

[arildno]

You did mean Leibniz' formula, HallsofIvy?

 

[HallsofIvy]

I am always making that mistake. Do you suppose I could convince them to swap names?