How do you do this question without looking at the answer choices?

Take a look at the 'proof' below

Step-by-step explanation:

The questions asks us to determine the anti-derivative of the function f(x) = 4x^3 sec^2 x^4. Let's start by converting this function into integral form. That would be the following:

Now all we have to do is solve the integral. Let's substitute 'u = x^4' into the equation 'du/dx = 4x^3.' We will receive dx = 1/4x^3 du. If we simplify a bit further:

Our hint tells us that d/dx tan(x) = sec^2(x). Similarly in this case our integral boils down to tan(u). If we undo the substitution, we will receive the expression tan(x^4). Therefore you are right, the first option is an anti-derivative of the function f(x) = 4x^3 sec^2 x^4.

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step-by-step explanation: