1. ## Solve the equaton

$\displaystyle \int\limits_{-\pi/2}^{\pi/2} sin^2x cos^2x (sinx+cosx)dx.$

2. Originally Posted by zorro
$\displaystyle \int\limits_{-\pi/2}^{\pi/2} sin^2x cos^2x (sinx+cosx)dx.$
$\displaystyle \sin^2{x} \cos^2{x} (\sin{x}+\cos{x}) =$

$\displaystyle \sin^2{x} \cos^2{x} \sin{x} + \sin^2{x} \cos^2{x} \cos{x} =$

$\displaystyle (1 - \cos^2{x}) \cos^2{x} \sin{x} + \sin^2{x}(1 - \sin^2{x})\cos{x} =$

$\displaystyle (\cos^2{x} - \cos^4{x})\sin{x} + (\sin^2{x} - \sin^4{x})\cos{x}$

$\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\cos^4{x} - \cos^2{x})(-\sin{x}) \, dx + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\sin^2{x} - \sin^4{x})\cos{x} \, dx$

for the first integral, let $\displaystyle u = \cos{x}$

for the second integral, let $\displaystyle u = \sin{x}$

3. Originally Posted by skeeter
$\displaystyle \sin^2{x} \cos^2{x} (\sin{x}+\cos{x}) =$

$\displaystyle \sin^2{x} \cos^2{x} \sin{x} + \sin^2{x} \cos^2{x} \cos{x} =$

$\displaystyle (1 - \cos^2{x}) \cos^2{x} \sin{x} + \sin^2{x}(1 - \sin^2{x})\cos{x} =$

$\displaystyle (\cos^2{x} - \cos^4{x})\sin{x} + (\sin^2{x} - \sin^4{x})\cos{x}$

$\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\cos^4{x} - \cos^2{x})(-\sin{x}) \, dx + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\sin^2{x} - \sin^4{x})\cos{x} \, dx$

for the first integral, let $\displaystyle u = \cos{x}$

for the second integral, let $\displaystyle u = \sin{x}$
how did u get a $\displaystyle - sinx$??????

4. Originally Posted by skeeter
$\displaystyle \sin^2{x} \cos^2{x} (\sin{x}+\cos{x}) =$

$\displaystyle \sin^2{x} \cos^2{x} \sin{x} + \sin^2{x} \cos^2{x} \cos{x} =$

$\displaystyle (1 - \cos^2{x}) \cos^2{x} \sin{x} + \sin^2{x}(1 - \sin^2{x})\cos{x} =$

$\displaystyle (\cos^2{x} - \cos^4{x})\sin{x} + (\sin^2{x} - \sin^4{x})\cos{x}$

$\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (\cos^4{x} - \cos^2{x})(-\sin{x}) \, dx + \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\sin^2{x} - \sin^4{x})\cos{x} \, dx$

for the first integral, let $\displaystyle u = \cos{x}$

for the second integral, let $\displaystyle u = \sin{x}$
$\displaystyle I_1 = \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} (cos^4 x - cos^2 x)(- sinx)dx$

u = cosx ; du = - sinxdx

$\displaystyle = \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} (u^4 - u^2 )du$

$\displaystyle \frac{1}{5} \left[ cos \left( - \frac{\pi}{2} \right) - cos \left( \frac{ \pi}{2} \right) \right]^5$ $\displaystyle - \frac{1}{3} \left[ cos \left( - \frac{\pi}{2} \right) - cos \left( \frac{ \pi}{2} \right) \right]^3$

= 0 ...........Is this correct ?????

Similarly

$\displaystyle I_2 = \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} (sin^2 x - sin^4 x)(cosx)dx$

v = sinx ; dv = cosxdx

$\displaystyle = - \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}} (u^2 - u^4 )du$

$\displaystyle - \frac{1}{3} \left[ sin \left( - \frac{\pi}{2} \right) - sin \left( \frac{ \pi}{2} \right) \right]^3$ $\displaystyle - \frac{1}{5} \left[ sin \left( - \frac{\pi}{2} \right) - sin \left( \frac{ \pi}{2} \right) \right]^5$

= $\displaystyle \frac{1}{3} (-1-1) - \frac{1}{5} (-1-1)$

= $\displaystyle -\frac{2}{3} + \frac{2}{5}$

= $\displaystyle - \frac{4}{15}$...............Is this correct????

5. Originally Posted by zorro
how did u get a $\displaystyle - sinx$??????
digging up the old posts, aren't you?

anyway ...

$\displaystyle (\cos^2{x} - \cos^4{x})\sin{x} = (\cos^4{x} - \cos^2{x})(-\sin{x})$

6. Originally Posted by skeeter
digging up the old posts, aren't you?

anyway ...

$\displaystyle (\cos^2{x} - \cos^4{x})\sin{x} = (\cos^4{x} - \cos^2{x})(-\sin{x})$
mite but is my work correct ????