# Math Help - Integral of sec^3 (theta) d(theta)

1. ## Integral of sec^3 (theta) d(theta)

How do I integrate sec^3 (theta) d(theta) ?

I am doing another problem and this is the step I am stuck on. I know all my previous steps are correct because math software told me but I'd like to be able to manually do this.

Any help would be greatly appreciated!

2. Originally Posted by s3a
How do I integrate sec^3 (theta) d(theta) ?

I am doing another problem and this is the step I am stuck on. I know all my previous steps are correct because math software told me but I'd like to be able to manually do this.

Any help would be greatly appreciated!
1)express sec as cos
2)multiply numerator and denominator by cos(theta)
3)denominator becomes an even power of cos. so express it as sin
4)substitute sin(theta)=t
5)use partial fractions
6)integrate

3. I subsitute for u instead of t because I am more familiar with it and I am at Integral of 1/(1-u^2)^2 du but am a bit lost now. Could you show me what to do with the partial fractions?

I don't know if the following is right:

1 = A(u+1)(u-1)^2 + B(u-1)^2 + C(u+1)^2 * (u-1) + D(u+1)^2

4. Originally Posted by s3a
How do I integrate sec^3 (theta) d(theta) ?
I always thought the method of parts was a neat way to do it ...

$\int \sec{t} \cdot \sec^2{t} \, dt$

$u = \sec{t}$ ... $du = \sec{t}\tan{t} \, dt$

$dv = \sec^2{t} \, dt$ ... $v = \tan{t}$

$\int \sec^3{t} \, dt = \sec{t}\tan{t} - \int \tan^2{t}\sec{t} \, dt$

$\int \sec^3{t} \, dt = \sec{t}\tan{t} - \int (sec^2{t}-1)\sec{t} \, dt$

$\int \sec^3{t} \, dt = \sec{t}\tan{t} - \int sec^3{t}-\sec{t} \, dt$

$\int \sec^3{t} \, dt = \sec{t}\tan{t} - \int sec^3{t} \, dt + \int \sec{t} \, dt$

$2\int \sec^3{t} \, dt = \sec{t}\tan{t} + \int \sec{t} \, dt$

$2\int \sec^3{t} \, dt = \sec{t}\tan{t} + \ln|\sec{t}+\tan{t}| + C
$

$\int \sec^3{t} \, dt = \frac{1}{2}\left[\sec{t}\tan{t} + \ln|\sec{t}+\tan{t}|\right] + C$

5. Just a question, what if I had a higher power though. Like what if I had Integral (sec^5(x))dx is there any easy method for that?