Hi! Thank you very much for your help---

I'm not sure what the answer to this is; how do I solve?

Find antiderivative of
(1/(x^2))[sec(1/x)][tan(1/x)]dx

I did integration by parts and got to
(1/(x^2))[sec(1/x)] + 2*[antiderivative of (1/(x^3))(sec(1/x))dx]

Integration by parts is the same as any other tool. It's just a tool. You can go around in circles with it... unless you know where you're going.

For this particular problem, I propose to use another tool, substitution.

Did you notice there is the factor (1/x²) at the beginning? What would ∫(1/x²)dx give? ∫-d(1/x).

So the integral becomes:
I=∫(1/(x^2))[sec(1/x)][tan(1/x)]dx
=∫[sec(1/x)][tan(1/x)]d(1/x)
=∫sec(y)tan(y)dy
= ... +C

Do remember, however, if and when you have to evaluate a definite integral, the limits have to correspond to the integration variable, which in this case is (1/x).

To find the antiderivative of the expression (1/(x^2))[sec(1/x)][tan(1/x)]dx, you have already made progress by using integration by parts.

Let's focus on the expression you've arrived at:

(1/(x^2))[sec(1/x)] + 2*[antiderivative of (1/(x^3))(sec(1/x))dx]

To simplify this further, let's break it down step by step:

Step 1: Evaluate the integral of (1/(x^2))[sec(1/x)].

The first part of the expression is already in a form that can be integrated easily. The integral of (1/(x^2)) can be found using the power rule for integration, which states that ∫(x^n)dx = (x^(n+1))/(n+1) + C, where C is the constant of integration.

Applying this rule, the antiderivative of (1/(x^2)) is -(1/x).

Therefore, the integral of (1/(x^2))[sec(1/x)] = -(1/x)[sec(1/x)] + C, where C is the constant of integration.

Step 2: Evaluate the integral of (1/(x^3))(sec(1/x)).

This part is a bit more complicated. To tackle this, you can use a substitution. Let u = 1/x, and then find du/dx by differentiating both sides of the equation:

du/dx = d/dx(1/x) = -1/(x^2)

Rearranging the equation, you get dx = -1/(u^2) du.

Now substitute u and dx in terms of u into the expression:

(1/(x^3))(sec(1/x)) dx = (1/(u^3))(sec(u)) (-1/(u^2) du)

Simplifying, you get -(1/u)(sec(u)) du.

The integral of -(1/u)(sec(u)) du can be evaluated by integration by substitution. Let v = sec(u). Using the same method as before, differentiate both sides to find dv/du:

dv/du = d/du(sec(u)) = sec(u)tan(u)

Rearranging, you get du = dv/(sec(u)tan(u)).

Substituting v and du in terms of v into the expression:

-(1/u)(sec(u)) du = -(1/u)(v) (dv/(sec(u)tan(u)))

Simplifying, you get -(1/u)(v)/(sec(u)tan(u)) dv.

Now, the expression becomes -(1/u)(v)/(sec(u)tan(u)) dv = -(1/u)(v)/(v) dv = -1/u dv.

After integrating -1/u with respect to v, you get -ln|v| + K, where K is the constant of integration.

Step 3: Combining the results from Steps 1 and 2.

Putting everything together, the final antiderivative is:

-(1/x)[sec(1/x)] - ln|sec(1/x)| + K, where K is the constant of integration.

This is the solution to the given integral.