Differentiate the following functions with respect to x.
a) [tex]\int[/tex] (t2 + cos(t7))/(1 + t4) dt [0, x]
b) [tex]\int[/tex] (t2 + cos(t7))/(1 + t4) dt [2, x]
c) [tex]\int[/tex] (t2 + cos(t7))/(1 + t4) dt [0, sin x]
d) [tex]\int[/tex] (t2 + cos(t7))/(1 + t4) dt [x, sin x]
e) [tex]\int[/tex] (x2 + cos(t7))/(1 + x4) dt [2, sin x]
Note: [a, b] means a is the lower limit of the integral and b is the upper limit of the integral.
I know that the derivative of an indefinite integral is the function itself and that with definite integrals you need to find an antiderivative G(x) and the derivative equals G(upper limit) - G(lower limit).
The Attempt at a Solution
I have no idea how to find an antiderivative for this function. Nor do I know how to use sin(x) as an upper limit. Nor do I know how to do (e) with x replacing some of the t's. Will x be considered a constant then, since we are differentiating in respect to t? I don't want the answers to all of them... I prefer to do the work on my own... That's how I actually learn, but if someone can give me a hint on how to start each one? I would be greatly appreciative!