I can't for the life of me evaluate this integral using the substitution method.

[The Student]

The indefinite integral is ∫(sec3x)^2 dx

So I am taught that we turn ∫f(g(x))(g'x)dx into ∫f(u)du. Practicing this, I turned ∫(sec3x)^2 dx into ∫f(u) du, where
u = sec(3x), and du = 3sec(3x)tan(3x)dx. Then sec(3x)tan(3x)dx = 3du. But there is no g'(x) or sec(3x)tan(3x) in the integrand for me to get to the next step. So unless I am suppose to put it outside of the integrand along with the 3, and I have tried that, I don't know what to do.

 

[srmichael]

The indefinite integral is ∫(sec3x)^2 dx

So I am taught that we turn ∫f(g(x))(g'x)dx into ∫f(u)du. Practicing this, I turned ∫(sec3x)^2 dx into ∫f(u) du, where
u = sec(3x), and du = 3sec(3x)tan(3x)dx. Then sec(3x)tan(3x)dx = 3du. But there is no g'(x) or sec(3x)tan(3x) in the integrand for me to get to the next step. So unless I am suppose to put it outside of the integrand along with the 3, and I have tried that, I don't know what to do.

You are overthinking this as is easy to do in calculus. This is one of those "canned" integrals that you need to have memorized (i.e. ∫sec²x dx = ?)

Here's a hint: What is the derivative of tan(x)?

 

[HallsofIvy]

The substitution you need is u= 3x.

 

[The Student]

The substitution you need is u= 3x.

I see. But how are we suppose to know to make u = 3x and not u = sec3x? I don't seem to be breaking any rules by having u = sec3x - or am I?

I worry about knowing the exact rules because on a test if I pick the wrong u, then that could mean a wrong answer.

 

[The Student]

You are overthinking this as is easy to do in calculus. This is one of those "canned" integrals that you need to have memorized (i.e. ∫sec²x dx = ?)

Here's a hint: What is the derivative of tan(x)?

Yeah I know the answer now, but I wish I knew why I can't have u = sec3x instead of u = 3x. They both have outer functions; only one version works, and the other doesn't.

 

[HallsofIvy]

Yeah I know the answer now, but I wish I knew why I can't have u = sec3x instead of u = 3x. They both have outer functions; only one version works, and the other doesn't.

You can't use u= sec(3x) because it leads to the problems you had in your first post. \(\displaystyle u= sec^2(3x)\) makes the integrand \(\displaystyle u^2\) but then \(\displaystyle du= 3sec(3x)tan(3x)dx\). You could divide both sideds by 3sec(3x)= 3u but you would still have that tan(3x)dx and there is no reasonable way to get rid of that tan(3x).

If, however you let u= 3x, du= 3dx so that (1/3)du= dx and the integral becomes \(\displaystyle \frac{1}{3}\int sec^2(u)du\). And that should be easy- you know that the derivative of tan(x) is \(\displaystyle sec^2(x)\) don't you?

 

[The Student]

You can't use u= sec(3x) because it leads to the problems you had in your first post. \(\displaystyle u= sec^2(3x)\) makes the integrand \(\displaystyle u^2\) but then \(\displaystyle du= 3sec(3x)tan(3x)dx\). You could divide both sideds by 3sec(3x)= 3u but you would still have that tan(3x)dx and there is no reasonable way to get rid of that tan(3x).

If, however you let u= 3x, du= 3dx so that (1/3)du= dx and the integral becomes \(\displaystyle \frac{1}{3}\int sec^2(u)du\). And that should be easy- you know that the derivative of tan(x) is \(\displaystyle sec^2(x)\) don't you?

Yes, I totally understand how having u = 3x works. I am just frustrated because I wish I knew why the process that we learnt breaks down when u = sec3x and f(u) = (u)^2. All of the other questions that we were assigned work when u = everything except for a single function, but in this case it doesn't work that way. When ever I come into one of these situations, it usually means that I am not understanding the process properly and then comes back to bite me in the ***. What in the general process of evaluating integrals says that I should not have even tried to make u = sec3x?

 

[HallsofIvy]

Yes, I totally understand how having u = 3x works. I am just frustrated because I wish I knew why the process that we learnt breaks down when u = sec3x and f(u) = (u)^2. All of the other questions that we were assigned work when u = everything except for a single function, but in this case it doesn't work that way. When ever I come into one of these situations, it usually means that I am not understanding the process properly and then comes back to bite me in the ***. What in the general process of evaluating integrals says that I should not have even tried to make u = sec3x?

No one is saying you "should not have even tried"! That is the process- you try something and see if it works. If it doesn't, try something else. And, except for the fact that these are class "exercises" that have been set up so the integral can be done, you can't really expect anything to work! Most functions do NOT have integrals in terms of elementary functions.