How Do You Integrate tan(x)sec^2(x)?

In summary, I am having trouble integrating tanxsec^2x. I tried doing something with the chain rule but it didn't work out at all. I know that I am supposed to show my work, but at this point I don't have anything to show you guys. Thanks a lot for any help you can give me.
  • #1
scorpa
367
1
Hello everyone,

I am having some trouble finding the integral of tanxsec^2x. I honestly have no idea where to go with this one. I've finished all the others but this one is really screwing me up. I tried doing something with the chain rule but it didn't work out at all. I know that I am supposed to show my work, but at this point I don't have anything to show you guys. Thanks a lot for any help you can give me.
 
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  • #2
scorpa said:
Hello everyone,

I am having some trouble finding the integral of tanxsec^2x. I honestly have no idea where to go with this one. I've finished all the others but this one is really screwing me up. I tried doing something with the chain rule but it didn't work out at all. I know that I am supposed to show my work, but at this point I don't have anything to show you guys. Thanks a lot for any help you can give me.

Write everything in terms of sines and cosines. Then you can do a simple substitution.
 
  • #3
Or just note that the derivative of tangent is the secant squared... u-substitution, anybody?

--J
 
  • #4
Ok guys, thanks for the help, but I'm still completely lost. I still don't have a clue of what I should do.
 
  • #5
Are you familiar with u-substitution? You're going to have to make one to evaluate the integral. You have two choices for what to let u equal, both work. What do you think u might equal?

--J
 
  • #6
I am sort of familiar with it. Could you let u equal tanx?
 
  • #7
scorpa said:
I am sort of familiar with it. Could you let u equal tanx?

If you did, what would du be?
 
  • #8
Ok, actually I might let u = sec^2x, then du should equal tanx+c?
 
  • #9
scorpa said:
Ok, actually I might let u = sec^2x, then du should equal tanx+c?

That is not correct

You should review the derivatives of all the trig functions.
 
  • #10
You're integrating u to get du for some reason. You must differentiate u to get du.

Stick with your original substitution.

--J
 
  • #11
[tex] \int \sec^{2} x \tan x \ dx=-\int \frac{d {}\cos x}{\cos^{3} x}=\frac{1}{2}\sec^{2}x +\mathcal{C} [/tex]

Daniel.
 
  • #12
shouldnt it be:
[tex]\int sec^2x~tanx~dx=\int\frac{sinx}{cos^3x}dx[/tex]
then, you could do u-sub and set [itex]u=cosx[/itex] and go from there?

*Edit*
dextercioby, you put sinx instead of cosx. i think tanx = sinx/cosx
*Edit*
 
Last edited:
  • #13
p53ud0 dr34m5 said:
shouldnt it be:
[tex]\int sec^2x~tanx~dx=\int\frac{sinx}{cos^3x}dx[/tex]
then, you could do u-sub and set [itex]u=cosx[/itex] and go from there?

and du would be

[tex]du = d cosx = -sinx dx[/tex]

giving

[tex] sinx dx = -d cosx[/tex]

which is what Dexter did to get his result, and what earlier hints were pointing to. It could also be done using

[tex] u= tanx [/tex]

[tex]du= d tanx = sec^2x dx[/tex]

[tex]sec^2x dx = d tanx[/tex]

as suggested by Justin early on
 
  • #14
Oh ok...I think I am starting to get it
 

Related to How Do You Integrate tan(x)sec^2(x)?

What is an integral of a trigonometric function?

An integral of a trigonometric function is a mathematical technique used to find the area under a curve of a trigonometric function. It involves finding the antiderivative of the function and evaluating it at the upper and lower limits of integration.

What are the basic trigonometric functions?

The basic trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent. These functions relate the angles of a right triangle to the lengths of its sides.

How do you integrate a sine function?

To integrate a sine function, you can use the formula ∫sin(x)dx = -cos(x) + C, where C is the constant of integration. This formula can be derived using the power rule for integration and the derivative of cosine.

Can the Pythagorean identities be used to integrate trigonometric functions?

Yes, the Pythagorean identities, such as sin^2(x) + cos^2(x) = 1, can be used to simplify integrals of trigonometric functions. They can also be used to convert between different trigonometric functions, making integration easier.

What are some common applications of integrals of trigonometric functions?

Integrals of trigonometric functions are commonly used in physics, engineering, and other fields to calculate the work done by a force, the displacement of an object, or the motion of a particle. They are also used in calculus to find the volume of irregular shapes and the length of curves.

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