Easy looks integral. Help

  • Thread starter alba_ei
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  • #1
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Hello I have this integral and I can't solve it


[tex]\int\csc^6 x dx[/tex]


Well i first start with trig identnity [tex]\csc x = \frac{1}{sen x}[/tex] so my first attempt to slove it looks like this

[tex]\int\frac{dx}{(\sin^2 x)^3}[/tex]

then i subsituted with [tex]\sin^2 x = \frac{1 - \cos 2x}{2}[/tex]

and i get

[tex]8\int\frac{dx}{(1-cos 2x)^3} [/tex] and i saw that i was going to

nowhere so i try another way

i took the function and use the trig identity to get this

[tex]\int\csc^6 x dx[/tex] =[tex]\int \frac{\sec^6 x}{\tan^6 x} dx [/tex]

[tex]u = \tan x[/tex] [tex]du = \sec^2 x[/tex]

then

[tex]\int\frac{du^3}{tan^6 x}[/tex]

and i get stucked again. any ideas?
 
Last edited:

Answers and Replies

  • #2
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i took the function and use the trig identity to get this

[tex]\int\csc^6 x dx[/tex] =[tex]\int \frac{\sec^6 x}{\tan^6 x} dx [/tex]
That looks most promising.

If [tex]u = \tan{x}[/tex], then [tex]du = \sec^2{x}dx[/tex] (Don't forget the dx)

Now, rewrite the integral as

[tex]\int{\frac{\sec^4{x}\sec^2{x}dx}{\tan^6{x}}}[/tex]

Now, think of an identity that relates [tex]\sec{x}[/tex] and [tex]\tan{x}[/tex].
 
Last edited:
  • #3
dextercioby
Science Advisor
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Insights Author
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Make [itex] \tan x =t [/itex] and then u'll get 3 simple integrals.
 
  • #4
VietDao29
Homework Helper
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Hello I have this integral and I can't solve it


[tex]\int\csc^6 x dx[/tex]
Here's the trick, if the power of csc function is even, you can use the u-substitution u = cot(x).
And if the power of sec function is even, you can use the u-substitution u = tan(x)
So, in your problem, the power of csc function is even, we use the u-substitution: u = cot(x)
du = -csc2(x) dx, so your integral will become:
[tex]\int \csc ^ 6 (x) dx = \int ( \csc ^ 4 (x) \times \csc ^ 2 (x) ) dx = \int {( \csc ^ 2 (x) )} ^ 2 \times \csc ^ 2 (x) dx = \int {\left( 1 + \cot ^ 2 (x) \right)} ^ 2 \times \csc ^ 2 (x) dx[/tex]
[tex]= -\int {(1 + u ^ 2)} ^ 2 du = ...[/tex]
Can you go from here? :)
 

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