# Math Help - How to integrate dx/(5+4sin x)

1. ## How to integrate dx/(5+4sin x)

$dx/(5+4sin x)$
step by step explanation would be appreciated
thanks

2. When you don't have any idea you can put $t=\tan \dfrac x2$.

3. yeah i took $t=tan (x/2)$ and $dt=sec²( x/2)$

But i don't know what to do after that :/

4. The complete substitution is...

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

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

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

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

Kind regards

$\chi$ $\sigma$

5. how did that sin x = 2t/(1+t²) thing come here?i don't know about that trigonometry identity.somebody want to prove it to me?

6. $\displaystyle \sin x = 2 \sin \frac{x}{2}\ \cos \frac{x}{2} = \frac{2 \sin \frac{x}{2}\ \cos \frac{x}{2}}{\sin^{2} \frac{x}{2} + \cos^{2} \frac{x}{2}} = \frac{2\ \tan \frac{x}{2}}{1+\tan^{2} \frac{x}{2}} = \frac{2 t}{1+t^{2}}$

Kind regards

$\chi$ $\sigma$

7. i know that [maths]sin (2x) =2 sin x.cos x [/maths].
but how is it $sin x =2 sin (x/2 ).cos (x/2)$?

and $[(2 sin (x/2) cos (x/2)]/[(sin² (x/2) + cos² (x/2)]= 2 Tan (x/2)/(1 + Tan² (x/2)$ ?

8. Originally Posted by silvercats
i know that [maths]sin (2x) =2 sin x.cos x [/maths].
but how is it $sin x =2 sin (x/2 ).cos (x/2)$?

and $[(2 sin (x/2) cos (x/2)]/[(sin² (x/2) + cos² (x/2)]= 2 Tan (x/2)/(1 + Tan² (x/2)$ ?
$sin(2x) = 2~sin(x)~cos(x)$

Now let y = 2x, Thus x = y/2...

For the other question, divide top and bottom by $cos^2 \left ( \frac{x}{2} \right )$

-Dan

9. thanks chisigma and topsquark !!!!!!