I don't understand how to do this one integral problem that involves secant. I'm asked to find the integral of sec^4 (4x). I'm not really sure how to go about solving this problem.

recall that sec^2 = 1+tan^2, so you have

∫sec^4(4x) dx
= ∫sec^2(4x)(1 + tan^2(4x)) dx
= ∫sec^2(4x) dx + ∫tan^2(4x) sec^2(4x) dx
= 1/4 tan(4x) + (1/4)(1/3) tan^3(4x)

and you can massage that in several ways.

To solve the integral of sec^4(4x), you can use a combination of trigonometric identities and integration techniques. Here's a step-by-step explanation of how to approach this problem:

Step 1: Rewrite sec^4(4x) in terms of cosine.
Using the identity sec^2(x) = 1 + tan^2(x), we can rewrite sec^4(4x) as (1 + tan^2(4x))^2.

Step 2: Use a substitution.
Let u = tan(4x). This substitution will simplify the integral and allow us to use standard integration techniques.

Step 3: Find du.
To find du/dx, differentiate u = tan(4x) with respect to x. Applying the chain rule, we have du/dx = 4sec^2(4x).

Step 4: Rearrange the equation to solve for dx.
Rearranging du/dx = 4sec^2(4x), we get dx = du / (4sec^2(4x)).

Step 5: Substitute u and dx in the integral.
Substituting u for tan(4x) and dx for du / (4sec^2(4x)), the integral becomes:
∫ (1 + u^2)^2 du / (4sec^2(4x))

Step 6: Simplify the integral.
Using the identity sec^2(x) = 1 + tan^2(x), we can rewrite sec^2(4x) as 1 + tan^2(4x). The integral now becomes:
∫ (1 + u^2)^2 du / (4(1 + u^2))

Step 7: Expand and simplify the integrand.
Expanding (1 + u^2)^2, we have ∫ (1 + 2u^2 + u^4) du / (4(1 + u^2)).
Simplifying this further, we get ∫ (1/4 + u^2/2 + u^4/4) du / (1 + u^2).

Step 8: Integrate term by term.
Integrating term by term, we have:
(1/4)∫ du / (1 + u^2) + (1/2)∫ u^2 du / (1 + u^2) + (1/4)∫ u^4 du / (1 + u^2).

Step 9: Solve each integral.
The first integral is the arctan function: (1/4)arctan(u).
The second integral can be simplified by substitution. Let z = 1 + u^2. Then, dz = 2u du, and the integral becomes:
(1/2)∫ dz / z = (1/2)ln|z| + C = (1/2)ln|1 + u^2| + C1.
The third integral also requires substitution. Let v = u^2. Then, dv = 2u du, and the integral becomes:
(1/4)∫ v^2 dv = (1/4)(v^3/3) + C2 = (1/4)(u^2)^3/3 + C2 = (1/12)u^6 + C2.

Step 10: Write final answer.
Combining the results from each integral, the final answer is:
(1/4)arctan(u) + (1/2)ln|1 + u^2| + (1/12)u^6 + C,
where C, C1, and C2 are constants of integration.

Step 11: Substitute back the original variable.
Substituting u = tan(4x) back into the answer will give the final result in terms of x.