Integrals and Derivatives: Solving Problems with Limits and Rates of Change

[SA32]

Homework Statement

In the attachment: 5aef, 6abc

The Attempt at a Solution

5a.) I started by dividing the question into two integrals: the integral of 2sin(x) evaluated from 0 to pi/4 minus the integral of 4sec²(x) evaluated from 0 to pi/4. Then the anti-derivative of 2sin(x) is -2cos(x) and the anti-derivative of 4sec²(x) is 4tan(x). I'm not sure if this is correct, though?

5e&f.) I'm not even sure what this is asking. It looks like "the integral of the derivative", which, according to my notes, is f(x)+C, but... help? I'm not sure.

6abc.) Again, not sure what this is asking. Looks like the derivative of the integral, which is f(x) in my notes. I made an attempt at the first one using the way my friend said to do it, you can see it in the second attachment, but I don't know if it is correct.

Thanks for any help!

 

[daveb]

Yes, 5e and 5f are asking for the integral of a derivative, and 6 is asking for the derivative of an integral.

 

[dextercioby]

For 5a) what you're thinking is right so keep doing it.

For 5e) or 5f) use that

$$\int_{a}^{b} \frac{df}{dx}{}dx= f(b)-f(a)$$

For 6abc) use the Leibniz-Newton formula and the chain rule of differentiation.

Daniel.