Trouble with trig identities in solving integral.

In summary: \displaystyle =\frac{1}{\cos^2(x) }\frac {\cos^2(x)}{\sin^2(x)}\frac {\cos^2(x)}{\sin^2(x)}\displaystyle =\frac{1}{\cos^2(x) }\frac {\cos^2(x)}{\sin^2(x)}\frac {\cos^2(x)}{\sin^2(x)}\displaystyle =\frac{1}{\cos^2(x)}\frac {\cos^2(x)}{\sin^2(x)}\frac {\cos^2(x)}{\sin^2(x)}\displaystyle =\frac{1}{\cos^2(x)}\frac {\cos
  • #1
NewtonianAlch
453
0

Homework Statement


[itex]\int[/itex][itex]\frac{sec^{2}x}{tan^{4}x}[/itex]dx

Homework Equations


The Attempt at a Solution



I have the answer as -1/3 cot^3(x) + C listed.

All the intermediate steps are given, but the first one is they have converted the sec/tan integral in to:

[itex]\int[/itex]{cot^{2}x}{csc^{2}x} dx

I am a little confused by the trig identities used and manipulated to get this.
 
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  • #2
you can just write it all in terms of sines and cosines and then reduce it to get that.
without using any identities
 
  • #3
Hint: what's the derivative of [itex]tan(x)[/itex]?
 
  • #4
NewtonianAlch said:

Homework Statement



[tex]\int\frac{\sec^{2}x}{\tan^{4}x} dx[/tex]
...

I am a little confused by the trig identities used and manipulated to get this.

[tex]\frac{\sec^{2}(x)}{\tan^{4}(x)}=\frac{1}{\cos^2(x)} \frac {\cos^4(x)}{\sin^4(x)}[/tex]

Try it from there.
 
  • #5
Following TheoMcCloskey's hint, the problem can be done without converting to sines and cosines. After a fairly simple substitution, the integral looks like
[tex]\int u^{-4}du[/tex]
 
  • #6
Thanks for the responses guys, yes the d/dx (tan x) substitution can be done to get a result of

-1/3*1/tan^3 x + C

But there's that other result which is bugging me, I will take a closer look at the post with the other identities as I want to see how I can get the cot^2csc^2 result.
 
  • #7
Taking it one step further ...

[itex]\displaystyle \frac{\sec^{2}(x)}{\tan^{4}(x)}=\frac{1}{\cos^2(x) } \frac {\cos^4(x)}{\sin^4(x)}[/itex]
[itex]\displaystyle =\frac{1}{\cos^2(x) }\frac {\cos^2(x)}{\sin^2(x)}\frac {\cos^2(x)}{\sin^2(x)}[/itex]​
 

Related to Trouble with trig identities in solving integral.

1. What are common mistakes when solving integrals involving trig identities?

Some common mistakes when solving integrals involving trig identities include not simplifying the expression before applying the identity, using the wrong identity, and forgetting to account for the limits of integration.

2. How do I know which trig identity to use when solving an integral?

The best way to determine which trig identity to use is to first simplify the expression as much as possible. Then, look for patterns or relationships between the trig functions in the simplified expression. This can help you identify which identity to use.

3. Can I use multiple trig identities in one integral?

Yes, it is possible to use multiple trig identities in one integral. However, it is important to be careful and keep track of which identity is being applied at each step to avoid making mistakes.

4. How do I check if my solution to an integral involving trig identities is correct?

You can check your solution by differentiating it. If the result is the original integrand, then your solution is correct. You can also use an online integral calculator to verify your solution.

5. Are there any tips for becoming better at solving integrals involving trig identities?

Practice and familiarize yourself with the common trig identities. It can also be helpful to look for patterns and relationships among the trig functions in the integrand. Additionally, try simplifying the expression as much as possible before applying the identity. And always remember to check your solution to ensure accuracy.

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