Integral Question: ∫(sec^2 x tanx) dx from 0 to ∏/3

[LocalStudent]

Homework Statement

∫(sec^2 x tanx) dx

the integral is from 0 to ∏/3

Homework Equations

I tried using integration by parts

The Attempt at a Solution

When I did integration by parts I got to this:

tan^2 x - ∫(sec^2 x tanx) dx (integral also from 0 to ∏/3)

which take you back to the question I started with.

 

[Fightfish]

Hint: Are there trigonometric identities that relate $sec^{2}x$ and $tan^{2}x$?

 

[Jorriss]

When I did integration by parts I got to this:

tan^2 x - ∫(sec^2 x tanx) dx (integral also from 0 to ∏/3)

which take you back to the question I started with.

There is a direct u-sub to handle this integral, but the way you have done it does illustrate an important concept.

Suppose, I = ∫Adx and you perform integration by parts and get I = B - ∫Adx. Well, that second integral is, as you noted, what you started with. Thus, I = B - I and so 2I = B and thus, I=B/2.

That type of oscillatory behavior is very important for many trigonometric and exponential integrals.

 

[LocalStudent]

Suppose, I = ∫Adx and you perform integration by parts and get I = B - ∫Adx. Well, that second integral is, as you noted, what you started with. Thus, I = B - I and so 2I = B and thus, I=B/2.
integrals.

Thanks, that was really helpful. I hope I remember and notice it in my test.

 

[LocalStudent]

There is a direct u-sub to handle this integral, but the way you have done it does illustrate an important concept.

Suppose, I = ∫Adx and you perform integration by parts and get I = B - ∫Adx. Well, that second integral is, as you noted, what you started with. Thus, I = B - I and so 2I = B and thus, I=B/2.

That type of oscillatory behavior is very important for many trigonometric and exponential integrals.

Thanks again! This exact question came up in my math test :)