Integration help! Integrate tan(x)sec^2(x)e^tan(x)

[aero2146]

Integrate tan(x)sec^2(x)e^tan(x)

This is something that I have to integrate to solve: cos^2(x) dy/dx + y = tan(x)

I divided by cos^2(x) on both sides and ended up with:
dy/dx + y*sec^2(x) = tan(x)sec^2(x)

Now I used the integrating factor:
h(x) = e^integral(sec^2(x)) = e^tan(x)

Then multiply by h(x) on both sides:
dy/dx*e^tan(x) + y*sec^2(x)e^tan(x) = tan(x)sec^2(x)e^tan(x)

Now simplifying it:
d/dx(e^tan(x)*y) = integral(tan(x)sec^2(x)e^tan(x))

This is where I don't understand how to integrate. I tried parts, but it doesn't seem like I'm going anywhere with it, because the power of tan just increases.

P.S: Can someone link me where to learn to type in Latex?

 

[Deleted member 4993]

Integrate tan(x)sec^2(x)e^tan(x)

This is something that I have to integrate to solve: cos^2(x) dy/dx + y = tan(x)

I divided by cos^2(x) on both sides and ended up with:
dy/dx + y*sec^2(x) = tan(x)sec^2(x)

Now I used the integrating factor:
h(x) = e^integral(sec^2(x)) = e^tan(x)

Then multiply by h(x) on both sides:
dy/dx*e^tan(x) + y*sec^2(x)e^tan(x) = tan(x)sec^2(x)e^tan(x)

Now simplifying it:
d/dx(e^tan(x)*y) = integral(tan(x)sec^2(x)e^tan(x))

This is where I don't understand how to integrate. I tried parts, but it doesn't seem like I'm going anywhere with it, because the power of tan just increases.

P.S: Can someone link me where to learn to type in Latex?

\(\displaystyle \displaystyle{\int tan(x) * sec^2(x) * e^{tan(x)} dx}\)

substitute

u = tan(x) → du = sec²(x) dx

continue.....

 

[aero2146]

\(\displaystyle \displaystyle{\int tan(x) * sec^2(x) * e^{tan(x)} dx}\)

substitute

u = tan(x) → du = sec²(x) dx

continue.....

Can't believe I didn't see that. Thanks for the help!

 

[Ishuda]

...

P.S: Can someone link me where to learn to type in Latex?

For this forum, you surround the LaTex with the [ tex ] [/ tex] tag pair without the spaces. A good starting point for the LaTex itself is, IMO,
ftp://ftp.ams.org/pub/tex/doc/amsmath/short-math-guide.pdf

As an example

[tex]\int_0^1 x\, dx[/tex]

will produce
\(\displaystyle \int_0^1 x\, dx\)
The \, is a space.

For other examples, just do a Reply with Quote on some of the posts