Find Integral of sin(x) / cos(x)^2 - Homework Help

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In summary, the conversation discusses methods for finding the integral of sin(x) / cos(x)^2, including using integration by parts and substitution. The final solution involves substituting u = cos(x)^2 and using standard integration techniques.
  • #1
ns5032
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Homework Statement



I would just like some help on how to find the integral of:
sin(x) / cos(x)^2



Homework Equations



integration by parts?
uv-integral of v(du)


The Attempt at a Solution


I tried using integration of parts with u = cos(x)^2 and dv = sin(x) but I just got myself in a bigger mess.

I noticed that sin(x) / cos(x)^2 was equal to tan(x)sec(x). Not sure if that helps though...

Help please?!
 
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  • #2
I'd go for a substitution.
 
  • #3
Try the substitution cos(x)^2.
 
  • #4
what's integral of [(-2*u)/(1-u^2)]/-2

or

try to see 1/x connection
 
Last edited:
  • #5
[tex]\int \frac{sinx}{cos^2x} dx[/tex]

[tex]\equiv \int \frac{sinx}{cosx} \times \frac{1}{cosx}dx[/tex]

What is another way to write [itex]\frac{sinx}{cosx}[/itex]?
and similarly [itex]\frac{1}{cosx}[/itex]?


Should be pretty standard after you see it.
 
  • #6
Got it. Thank you everyone.
 

1. What is the integral of sin(x) / cos(x)^2?

The integral of sin(x) / cos(x)^2 is -cot(x) + C, where C is the constant of integration.

2. How do I solve this type of integral?

To solve this type of integral, you can use the trigonometric identity cos^2(x) = 1 - sin^2(x) and then substitute u = sin(x) and du = cos(x)dx. This will transform the integral into -1/u^2 du, which can be easily integrated.

3. Can I use a different substitution to solve this integral?

Yes, there are multiple ways to solve integrals. You can also use the substitution u = cos(x) or u = tan(x) to solve this integral.

4. Is there a shortcut or trick to solving this integral?

Yes, you can use the trigonometric identity cos^2(x) = 1 - sin^2(x) to simplify the integral and make it easier to solve. You can also use a different substitution to make the integral more manageable.

5. Can you provide a step-by-step solution for this integral?

Sure, first use the substitution u = sin(x) and du = cos(x)dx, which transforms the integral into -1/u^2 du. Then, integrate -1/u^2 to get -1/u. Finally, substitute back u = sin(x) and add the constant of integration to get the final solution of -cot(x) + C.

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