# Interesting Integral

• Jun 14th 2011, 11:01 PM
watchmath
Interesting Integral
Compute
$\int_0^{\pi/2}\frac{\sin x}{9+16\sin(2x)}\,dx$
• Jun 14th 2011, 11:12 PM
chisigma
Re: Interesting Integral
Use the substitutions...

$t= \tan \frac{x}{2}$

$dx = 2\ \frac{dt}{1+t^{2}}$

$\sin x = \frac{2 t}{1+t^{2}}$

$\cos x = \frac{1-t^{2}}{1+t^{2}}$

... and take into account the identity...

$\sin 2x = 2\ \sin x\ \cos x$

Kind regards

$\chi$ $\sigma$
• Jun 14th 2011, 11:29 PM
watchmath
Re: Interesting Integral
I am not sure with that. The rational function will be very ugly ...
• Jun 15th 2011, 07:27 AM
topsquark
Re: Interesting Integral
Quote:

Originally Posted by watchmath
I am not sure with that. The rational function will be very ugly ...

Not so bad, though, from a Calculus standpoint. Notice that the denominator is a biquadratic: $at^4 + bt^2 + c$. Use a substitution u = t^2 and the form will be much more recognizable.

-Dan
• Jun 15th 2011, 11:06 PM
watchmath
Re: Interesting Integral
I got
$\frac{4t}{9t^4-64t^3+18t^2+64t+9}$
and I would say that is ugly :).
• Jun 16th 2011, 02:19 AM
Prove It
Re: Interesting Integral
Quote:

Originally Posted by watchmath
I got
$\frac{4t}{9t^4-64t^3+18t^2+64t+9}$
and I would say that is ugly :).

The bottom factorises to $\displaystyle (t^2 - 8t + 9)(9t^2 + 8t + 1)$, so the next step is partial fractions.
• Jun 16th 2011, 02:30 AM
TheCoffeeMachine
Re: Interesting Integral
Quote:

Originally Posted by Prove It
The bottom factorises to $\displaystyle (t^2 - 8t + 9)(9t^2 + 8t + 1)$, so the next step is partial fractions.

How did you obtain this factorisation, if you don't mind me asking? I'm not seeing it; neither can I find a clever/painless way of solving the integral! :(
• Jun 16th 2011, 02:37 AM
Prove It
Re: Interesting Integral
Quote:

Originally Posted by TheCoffeeMachine
How did you obtain this factorisation, if you don't mind me asking? I'm not seeing it; neither can I find a clever/painless way of solving the integral! :(

I used Wolfram Alpha, lol...
• Jun 16th 2011, 03:08 AM
bugatti79
Re: Interesting Integral
Quote:

Originally Posted by Prove It
I used Wolfram Alpha, lol...

(Tongueout) lol!!