# Improper Integral of theta/cos^2 theta

• leo255
In summary, the conversation is about a student struggling with an improper integral question involving theta and cosine squared. They attempt to solve it using trigonometric identities and integration by parts, but realize their answer is incorrect. The conversation also touches on the concept of improper integration and the properties of logarithmic functions.

## Homework Statement

Improper Integral of theta/cos^2 theta

## The Attempt at a Solution

Hi all, this was one of the few questions on my final today that I just didn't know how to do. I know how to do trig sub, know all my trig identities and know improper integration, but was a bit at a loss for this one.

I could use a half angle for the denominator --> theta / 1/2 [1 + cos(2 theta)] -->

Maybe integrate, and get theta^2 / 1/2 theta + 1/2(sin 2 theta).

I'm sure what I tried was very wrong, but I wanted to make some kind of attempt.

Edit: nevermind, you can't integrate like that.

Last edited:
leo255 said:

## Homework Statement

Indefinite Integral of theta/cos^2 theta

## The Attempt at a Solution

Hi all, this was one of the few questions on my final today that I just didn't know how to do. I know how to do trig sub, know all my trig identities and know improper integration, but was a bit at a loss for this one.

I could use a half angle for the denominator --> theta / 1/2 [1 + cos(2 theta)] -->

Maybe integrate, and get theta^2 / 1/2 theta + 1/2(sin 2 theta).

I'm sure what I tried was very wrong, but I wanted to make some kind of attempt.

Edit: nevermind, you can't integrate like that.
$$\int \frac{\theta d\theta}{cos^2(\theta)} = \int \theta sec^2(\theta) d\theta$$

Use integration by parts with a judicious choice for u and dv.

Damn, that's a pretty easy integration by parts question actually. So, if I get an answer of tan(theta) - ln(sec(theta)), where would the improper integration come into play?

Oh wait, natural log functions must be greater than zero. So, it would be something like, the limit, as b approaches 0, from the right, of tan(theta) - ln(sec(theta))?

leo255 said:
Damn, that's a pretty easy integration by parts question actually. So, if I get an answer of tan(theta) - ln(sec(theta)), where would the improper integration come into play?
It's not a hard integration by parts, but the answer you show is incorrect. If you differentiate your answer, you don't get ##\theta sec^2(\theta)##.
leo255 said:
Oh wait, natural log functions must be greater than zero. So, it would be something like, the limit, as b approaches 0, from the right, of tan(theta) - ln(sec(theta))?
To be more precise, the argument of a log function must be greater than zero. The output of a log function can be any real number.

The integral you showed was an indefinite integral. An improper integral is a definite integral for which the integrand is undefined at one or more points inside the interval defined by the limits of integration, or at one or both endpoints.