Need Integral Help (sec^3(x))

[rhm95]

I was given the problem to find the integral sec³(x). I have gotten some progress on this.

change sec³x -> 1/cos³x -> cosx/cos⁴x -> cosx/(1-sin²x)²

usub -> u=sinx du = cosx dx -> 1/(1-u²)² -> 1/(u⁴-2u²+1) -> 1/(u²-1)(u²+1) -> 1/(u+1)(u-1)(u+1)(u-1) -> 1/(u+1)²(u-1)²

partial fractions -> a/(u+1) + b/(u+1)²+ c/(u-1) + d/(u+1)<sup>2


I can solve for b and d and I get 1/4 and 1/4 respectively. I don't know how to proceed from there though. I cant solve for a and c.
Is this even the right approach? Or did I make a mistake somewhere?</sup>

 

[Deleted member 4993]

I was given the problem to find the integral sec³(x). I have gotten some progress on this.

change sec³x -> 1/cos³x -> cosx/cos⁴x -> cosx/(1-sin²x)²

usub -> u=sinx du = cosx dx -> 1/(1-u²)² -> 1/(u⁴-2u²+1) -> 1/(u²-1)(u²+1) -> 1/(u+1)(u-1)(u+1)(u-1) -> 1/(u+1)²(u-1)²

partial fractions -> a/(u+1) + b/(u+1)²+ c/(u-1) + d/(u+1)<sup>2


I can solve for b and d and I get 1/4 and 1/4 respectively. I don't know how to proceed from there though. I cant solve for a and c.
Is this even the right approach? Or did I make a mistake somewhere?</sup>

1/(1-u²)² = 1/4 * [1/(1-u) + 1/(1+u)]² =1/4 * [{1/(1-u)}² + {1/(1+u)}² + 2/{(1+u)(1-u)}] = 1/4 * [1/(1-u)² + 1/(1+u)² + 1/(1+u) +1/(1-u)] ... Now continue....

 

[rhm95]

1/(1-u²)² = 1/4 * [1/(1-u) + 1/(1+u)]²

How were you able to just split that up into two fractions? Also where did the 1/4 come from?

 

[Deleted member 4993]

How were you able to just split that up into two fractions? Also where did the 1/4 come from?

Because, I know how to factorize (1-u²) [=(1+u)(1-u)] and I know 1/(1+u) + 1/(1-u) → 2/(1-u²)

 

[rhm95]

1/4 * [1/(1-u)² + 1/(1+u)² + 1/(1+u) +1/(1-u)] ... Now continue....

So are you saying that this is what you end up with if do the partial fraction? All that needs to been done is to integrate the rest? If so, could you provide some clarification as to how you solved for that cause I'm still very confused.

 

[Deleted member 4993]

So are you saying that this is what you end up with if do the partial fraction? All that needs to been done is to integrate the rest? If so, could you provide some clarification as to how you solved for that cause I'm still very confused.

Are working with a pencil and paper or are you just staring at my post! This is simple algebra - if you work through it - it should be obvious!!! I have worked out ~75% for you.