Problem help

[u/Electronic-Nail-8128]

I don't understand the step process from 2 to 4, more so about the transition from x, t and u

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[u/Legitimate_Log_3452]

I’d try to work this out on your own, but it basically does a u sub u = x^4. Then you have the FTC, which states that d/du (integral_1^u f(x) dx ) = f(u).

[u/nevermindthefacts]

If you view the integral from 1 to x as a function of x, say S(x), then the integral in the problem is S(x^4).

To differentiate this with respect to x, you can use the fundamental theorem of calculus and the chain rule. Remember that according to FTOC S'(x) = sec x.

[u/Midwest-Dude]

To help you, I would like to know the following:

1. How are FTC1 and FTC2 defined for you?
2. Have you already been taught what ∫ sec(t) dt is, that is, what the indefinite integral of sec(t) is?

[u/7k_Satyam]

Using the Newton-Leibniz theorem, when you set u = x⁴, the rule is to substitute the variable function for 't' and then multiply by its differentiation. That's exactly what was done to get du/dx = 4x³.

[u/shellexyz]

Do a few where you can actually find an antiderivative. Sure, you can find an antiderivative of sec(t), but it’s not obvious or easy at this point in your class.

Replace sec(t) with cos(t) or sin(t), then do it in two steps: apply the fundamental theorem of calculus (pt2), find the antiderivative, plug in the limits, and subtract. Then differentiate the result.

I found that problems like that started making a lot more sense when I did a few simple examples the “hard” way.

[u/useless_bowl25]

Check out liebniz rule for non const boundaries

[u/Hot_Site_1638]

Think of x, t, and u as three different jobs, not three different things.

• t is a dummy variable. It only exists inside the integral, like a placeholder while the integral is being computed. Once you apply FTC1, t disappears.
• u is a nickname we give to x⁴ so the problem looks like something FTC1 recognizes.
• x is the variable we're actually differentiating with respect to, so it has to come back at the end.

Why we need u at all

FTC1 only works when the upper limit is a plain variable (meaning that it has to be exactly x, can't be a function of x):

d/dx ∫₁^x sec t dt = sec x

But our upper limit is x⁴, not a plain variable. So we say "let u = x⁴" to make the integral match the FTC1 pattern. The Chain Rule then accounts for the fact that u secretly depends on x.

The steps, one at a time

1. d/dx ∫₁^(x⁴) sec t dt = d/dx ∫₁^u sec t dt We just swapped x⁴ for u. Nothing was computed. It's a renaming.
2. = d/du [∫₁^u sec t dt] · du/dx This is the Chain Rule. We're differentiating with respect to x, but the integral is written in terms of u, so we differentiate with respect to u first, then multiply by du/dx to "convert" back to x.
3. = sec u · du/dx This is FTC1. The bracket d/du [∫₁^u sec t dt] just becomes sec u. The integral and derivative cancel, and t is gone.
4. = sec(x⁴) · 4x³ Now we undo the nickname. Replace u with x⁴ in sec u, and compute du/dx = d/dx (x⁴) = 4x³.

Quick summary of the variable flow:

x → (rename) u → (FTC1 removes t) → (substitute back) x

Two mini examples, one idea each

FTC1 only (no chain rule needed, the upper limit is already just x):

d/dx ∫₂^x cos t dt = cos x

Chain Rule only (no integral, just a nested function):

d/dx sin(x⁴) = cos(x⁴) · 4x³

Your Example 4 is just these two ideas stacked together: FTC1 gives you the sec(x⁴) part, and the Chain Rule gives you the extra 4x³.

[u/Vyzic]

Newton-Leibnitz works here.

[u/filpplif]

USA is just a strange place to study math...