# How to integrate (e^t)*tan^2(t)

1. Oct 21, 2012

### digipony

1. The problem statement, all variables and given/known data

I am in the process of solving second order differential equation by the variation of parameters method. I need to calculate ∫(et)*(tant)2 dt

2. Relevant equations
(tanx)2 = (secx)2-1
∫(secx)dx= lnlsecx*tanxl

3. The attempt at a solution
I have used the trig identity to get this: ∫(et*((sect)2-1)dt
= ∫et*(sect)2 dt -∫(et) dt
=∫et*(sect)2 dt -et
but am stuck here. Any help would be greatly appreciated. Thanks!

Last edited: Oct 21, 2012
2. Oct 21, 2012

### dextercioby

I think it's not expressible in terms of elementary functions. Find another method for your ODE.

3. Oct 21, 2012

### Dickfore

4. Oct 21, 2012

### digipony

The problem explicitly asks to solve using variation of parameters. Perhaps I made a mistake in the algebra somewhere before this step; I will go to my professor. Thanks for your input, now I won't waste a hour or two staring at and trying to solve this integral.

5. Oct 21, 2012

### dextercioby

State the ODE, please. There might be a mistake somewhere, or your professor really likes Gauss hypergeometric functions.

6. Oct 21, 2012

### digipony

I doubt it; I probably made a mistake. Here is the problem:

Find the general solution by variation of parameters:
y'' + y = (tanx)2

For these, I know you find the solution by finding and adding together the solution to the homogeneous equation with that of your particular solution.

I found the solution to the homogeneous equation to be:
yh = C1*e-t+C2

Then I needed to find the particular solution which I knew would be in the form u1*y1 + u2*y2
where
u1 = -∫ $\frac{y2*g}{W[y1,y2]}$ dt
and
u2= ∫ $\frac{y1*g}{W[y1,y2 ]}$ dt

(g is (tanx)2, and W is the Wronskian)

7. Oct 21, 2012

### dextercioby

It's not right. y''+y=0 doesn't have the solution you wrote as y_h. No way.

8. Oct 21, 2012

### digipony

Ah! I see. I was using reduction of power method and messed up. I had gotten r2 + r =0 whereas it should be r2+1=0 . Therefore I should get an imaginary root, and the solution is yh= C1*sint + C2*cost