How do you use part I of the Fundamental Theorem of Calculus to find the derivative of h(x) = int (cos(t^4) + t) dt from -4 to sinx? Can someone walk me through this? I'm having a lot of issues getting a grasp on how to do this.?

Jul 30, 2015

The answer is $h ' \left(x\right) = \left(\cos \left({\sin}^{4} \left(x\right)\right) + \sin \left(x\right)\right) \cdot \cos \left(x\right)$.

Explanation:

If you define a function $g$ by the formula $g \left(x\right) = {\int}_{- 4}^{x} \left(\cos \left({t}^{4}\right) + t\right) \setminus \mathrm{dt}$, then the Fundamental Theorem of Calculus says that its derivative is $g ' \left(x\right) = \cos \left({x}^{4}\right) + x$ (get rid of the integral sign and the $\mathrm{dt}$, and replace the $t$ in the integrand with $x$...the $- 4$ in the lower limit of the integral is irrelevant (it could be any number and the answer would be the same), but the $x$ in the upper limit of the integral is essential)

Now notice that $h \left(x\right) = {\int}_{- 4}^{\sin \left(x\right)} \left(\cos \left({t}^{4}\right) + t\right) \setminus \mathrm{dt} = g \left(\sin \left(x\right)\right)$ ($h$ is a composition of $g$ with the sine function).

You can now apply the Chain Rule to say that

$h ' \left(x\right) = g ' \left(\sin \left(x\right)\right) \cdot \frac{d}{\mathrm{dx}} \left(\sin \left(x\right)\right)$

$= \left(\cos \left({\sin}^{4} \left(x\right)\right) + \sin \left(x\right)\right) \cdot \cos \left(x\right)$

Perhaps there is still confusion about what $g$ and $h$ are. In other words, do they have "ordinary formulas" that don't involve integral signs. The answer, in this case, is "no". The integral $\int \cos \left({t}^{4}\right) \setminus \mathrm{dt}$ cannot be evaluated in terms of "elementary functions" (functions that you are "used to").

The definite integral symbol $h \left(x\right) = {\int}_{- 4}^{\sin \left(x\right)} \left(\cos \left({t}^{4}\right) + t\right) \setminus \mathrm{dt}$ most certainly defies a function because the integrand is continuous. For any $x$, you can always approximate the value of $h \left(x\right)$ by numerical integration (like Simpson's Rule).