Question

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

Answer 1

Take a look at the 'proof' below

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:

`\mathrm{\int \:4x^3sec^2x^4dx}`

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:

`\mathrm{\int \:\:sec^2\left(u\right)du}`

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.