∫ cos^5 (17x) dx, between b = pi/2 and a = -2pi Help!
F flakine Junior Member Joined Aug 24, 2005 Messages 78 Sep 7, 2006 #1 ∫ cos^5 (17x) dx, between b = pi/2 and a = -2pi Help!
G galactus Super Moderator Staff member Joined Sep 28, 2005 Messages 7,216 Sep 7, 2006 #2 Here's a nice little general formula for the integrals of cos5(nx)\displaystyle cos^{5}(nx)cos5(nx) \(\displaystyle \L\\\int{cos^{5}(nx)}dx=\frac{sin(nx)(3cos^{4}(nx)+4cos^{2}(nx)+8)}{15n}\) You have n=17 If you must integrate it all the long arduous way, you could start with the identity 1−sin2(17x)=cos2(17x)\displaystyle 1-sin^{2}(17x)=cos^{2}(17x)1−sin2(17x)=cos2(17x) Then you have: \(\displaystyle \L\\\int_{-2\pi}^{\frac{\pi}{2}}(1-sin^{2}(17x))^{2}cos(17x)dx\)
Here's a nice little general formula for the integrals of cos5(nx)\displaystyle cos^{5}(nx)cos5(nx) \(\displaystyle \L\\\int{cos^{5}(nx)}dx=\frac{sin(nx)(3cos^{4}(nx)+4cos^{2}(nx)+8)}{15n}\) You have n=17 If you must integrate it all the long arduous way, you could start with the identity 1−sin2(17x)=cos2(17x)\displaystyle 1-sin^{2}(17x)=cos^{2}(17x)1−sin2(17x)=cos2(17x) Then you have: \(\displaystyle \L\\\int_{-2\pi}^{\frac{\pi}{2}}(1-sin^{2}(17x))^{2}cos(17x)dx\)
tkhunny Moderator Staff member Joined Apr 12, 2005 Messages 11,339 Sep 7, 2006 #3 I'm rather fond of some symmetry exploitation. I assumed you meant -π\displaystyle \piπ/2, not -2π\displaystyle \piπ \(\displaystyle \L\,2*\int_{0}^{\frac{\pi}{34}}{cos^{5}(17x)}\,dx\) Unless you REALLY meant -2π\displaystyle \piπ, then it's just \(\displaystyle \L\,\int_{0}^{\frac{\pi}{34}}{cos^{5}(17x)}\,dx\) Of course, that doesn't simplify the antiderivative any. It certainly does simplify life in the numerical world.
I'm rather fond of some symmetry exploitation. I assumed you meant -π\displaystyle \piπ/2, not -2π\displaystyle \piπ \(\displaystyle \L\,2*\int_{0}^{\frac{\pi}{34}}{cos^{5}(17x)}\,dx\) Unless you REALLY meant -2π\displaystyle \piπ, then it's just \(\displaystyle \L\,\int_{0}^{\frac{\pi}{34}}{cos^{5}(17x)}\,dx\) Of course, that doesn't simplify the antiderivative any. It certainly does simplify life in the numerical world.