# Math Help - how to split this integral..

1. ## how to split this integral..

$
\int_{0}^{\frac{\pi}{2}}\left|\frac{sin(\frac{3\pi }{2}+x)}{(3+\cos^2 x)(3-\sin x)}\right| dx
$

the denominator is positive the numerator needs to be splited

sinx>0 when $0

$0<\frac{3\pi}{2}+x<\pi$
$-\frac{3\pi}{2}

what now ??
how to split the integral and cancel the absolute value??

2. So the argument of the sine in the numerator goes from $\frac{3 \pi}{2} + 0 = \frac{3 \pi}{2}$ to $\frac{3 \pi}{2} + \frac{\pi}{2} = 2\pi$

Sine(x) in this domain is negative. So you simply require a negative sign in front of the integral and remove the absolute value signs:

$\int_{0}^{\frac{\pi}{2}}-\frac{sin(\frac{3\pi}{2}+x)}{(3+\cos^2 x)(3-\sin x)} dx
$

3. thanks