How do you integrate \(\int \sec^{2n+1} x \, dx\)?

[nhrock3]

$$\intop_{-\pi/2}^{+\pi/2}\sqrt{(-6sin2t)^2+\sqrt{6}(cost)^2}dt$$

how to solve it?
i tried
by parts and it looks very bad and complicated

edit
sorry i copied the wrond integral
now it fine

 

[mateomy]

Did you try a U substitution first?

 

[nhrock3]

what substitution?

 

[mateomy]

A "u" substitution. Where you pick a section of your integral and substitute it with "u" and replace your limits of integration. Its not in your book?

 

[nhrock3]

ok ill take u=sint
so du=costdt
but i can't have dt inside the root
and i can't take it aside apart

 

[mateomy]

You're going to need to use a trig identity too. Sorry, I am doing this in my head.

 

[nhrock3]

what identity?

 

[mateomy]

Look at your first term -12sintcost it should be pretty obvious after that.

 

[nhrock3]

yes i know it wwas sin2t before
and i brke it into 2sintcost
what to do?

 

[nhrock3]

i did z=sint
and simplified it to
$$\intop_{-\pi/2}^{+\pi/2}\sqrt{36z^2+6}dz$$

what now?

 

[hunt_mat]

1) You also have to also change the limits of integration, I think they are -1 and 1 now.

2) Take out a factor of 36 from the integral

3) This should now be a standard integral. If you need more assistance, I would think of letting $$z=\sinh w/\sqrt{6}$$ and working from there.

 

[nhrock3]

i haven't studied hiporbolic functions
could you write it into normal substitution
?

 

[hunt_mat]

Okay, try $$z=\tan w/\sqrt{6}$$

 

[Mark44]

Look at your first term -12sintcost it should be pretty obvious after that.

Maybe I'm forgetting something, but it's not obvious to me what to do after that.

i did z=sint
and simplified it to
$$\intop_{-\pi/2}^{+\pi/2}\sqrt{36z^2+6}dz$$

what now?

I don't see how you got this from your substitution of z = sin(t). dz = cos(t)dt, and as you pointed out already, you don't have a factor of cos(t) outside the integral.

 

[nhrock3]

Okay, try $$z=\tan w/\sqrt{6}$$

i did it
i got dz/(cos w) type integral
but i can't substitute dz with a w variable
?

 

[Mark44]

i did z=sint
and simplified it to
$$\intop_{-\pi/2}^{+\pi/2}\sqrt{36z^2+6}dz$$

what now?

1) You also have to also change the limits of integration, I think they are -1 and 1 now.

2) Take out a factor of 36 from the integral

3) This should now be a standard integral. If you need more assistance, I would think of letting $$z=\sinh w/\sqrt{6}$$ and working from there.

I could be wrong, but in step 3, hunt_mat is taking the integral at the top at face value. I don't believe that this integral follows from the original integral of this thread.

 

[hunt_mat]

I could be wrong, but in step 3, hunt_mat is taking the integral at the top at face value. I don't believe that this integral follows from the original integral of this thread.

Correct I was.

 

[nhrock3]

he is following the right integral
it developed into this new one
https://www.physicsforums.com/showpost.php?p=3247073&postcount=10
and here i tried to solve it
https://www.physicsforums.com/showpost.php?p=3247106&postcount=15

 

[hunt_mat]

The integral boils down to integrating

$$\int\sqrt{1+x^{2}}dx$$

though.

 

[nhrock3]

ok i will try this thing tommorow
good night :)

 

[Mark44]

$$\intop_{-\pi/2}^{+\pi/2}\sqrt{(-6sin2t)^2+\sqrt{6}(cost)^2}dt$$

how to solve it?
i tried
by parts and it looks very bad and complicated

edit
sorry i copied the wrond integral
now it fine

Well, that makes a big difference. As I recall, you originally had this integral.

$$\int_{-\pi/2}^{+\pi/2}\sqrt{-6sin2t+\sqrt{6}cost}dt$$

 

[SammyS]

Yes Mark. That editing feature can jump up and bite you.

 

[nhrock3]

i got now an integral of dt/(cost)^3

what to do?

 

[hunt_mat]

I am unsure how to do this integral without using hyperbolic function, I used the following functions:

$$\begin{array}{rcl}<br /> \sinh x & = & \frac{e^{x}-e^{-x}}{2} \\<br /> \cosh x & = & \frac{e^{x}+e^{-x}}{2} \\<br /> \cosh^{2}x-\sinh^{2}x & \equiv & 1 \\<br /> \cosh 2x & = & \cosh^{2}x+\sinh^{2}x \\<br /> \sinh 2x & = & 2\sinh x\cosh x<br /> \end{array}$$

With the above the integral becomes simple. The first two are definitions of cosh and sinh, the rest may be derived from those definitions.

 

[nhrock3]

how it become simple?

 

[hunt_mat]

For the integral I posted:

$$\int\sqrt{1+x^{2}}dx$$

Use the substitution $$x=\sinh u$$ and what do you get?

 

[nhrock3]

i used x=tan t substitution
and i got a different integral
dt/(cos t)^3
how to solve it

 

[hunt_mat]

I am not sure you can do it that way anymore, as I said, I have been thinking and the only straightforward way of doing it is hyperbolic functions, I put everything that you need in a previous post.

 

[nhrock3]

i want to solve it my way
i got now an integral of dt/(cost)^3

what to do?

 

[hunt_mat]

The only thing that I can think of is use:

$$t=\tan\left(\frac{x}{2}\right)$$

I don't think that it will get you anywhere with this method, good luck though.