help with integral

**BobbyBear** · Jun12-09, 05:07 PM

Hello, I need help witht he following integral which has come up while trying to solve a differential equation.

$LaTeX Code: <BR><BR> \\int \\frac{e^t}{cos^2(5t)}sin(6t)dt<BR><BR>$

I'm not sure it can be integrated analytically, I 've tried to think of possible trigonometric manipulations and integrating by parts ... but not leading anywhere, any help would be appreciated.

**Cyosis** · Jun12-09, 05:20 PM

I am pretty sure a primitive of this function does not exist in closed form/elementary functions.

**Mute** · Jun12-09, 06:06 PM

Gradshteyn & Rhyzik states that indefinite integrals of the forms exp[ax] divided by sin(bx), cos(bx), sin^2(bx) or cos^2(bx) aren't expressible as a finite combination of elementary functions, so your integral is similarly hopeless, as it's a linear combination of such integrals (for complex a).

You might be able to get an infinite series for it, though.

Or, if you had constant limits it might be a doable integral.

**BobbyBear** · Jun12-09, 06:30 PM

Thank you, I've been breaking my head trying to find a closed form in terms of elementary functions, but wasn't able to come up with anything. I guess the only way then is to use Taylor expansions or the like and then integrate the terms of the resulting series.

No, I don't have constant limits: the integral came up in the process of trying to find the solution function of an ordinary differential equation.

Thank you for your help :)