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sin5xCos3x

先积化和差,再积分! sin5xcos3x=0.5sin(8x)+0.5sin(2x) ∫sin5xcos3x dx =∫0.5sin(8x)+0.5sin(2x)dx =(1/16)∫sin(8x)d(8x) + (1/4)∫sin(2x)d(2x) =-(1/16)cos(8x) - (1/4)cos(2x) + C

先积化和差,再积分! sin5xcos3x=0.5sin(8x)+0.5sin(2x) ∫sin5xcos3x dx =∫0.5sin(8x)+0.5sin(2x)dx =(1/16)∫sin(8x)d(8x) + (1/4)∫sin(2x)d(2x) =-(1/16)cos(8x) - (1/4)cos(2x) + C

∫(sinx)^3·(cosx)^5dx =-∫(sinx)^2·(cosx)^5d(cosx) =∫[(cosx)^2-1](cosx)^5d(cosx) =∫(cosx)^7d(cosx)-∫(cosx)^5d(cosx) =(1/8)(cosx)^8-(1/6)(cosx)^6+C。

y'=5cos5xcos3x-3sin5xsin3x y''=5(-5sin5xcos3x-3sin3xcos5x)-3(5cos5xsin3x+3cos3xsin5x) =-25sin5xcos3x-15sin3xcos5x-15sin3xcos5x-15sin5xcos3x =-40sin5xcos3x-30sin3xcos5x 没怎么算,不知对否。

sin2xcos3x =1/2[sin(2x+3x)+sin(2x-3x)] =1/2(sin5x-sinx)

(cos5xcos3x)' =(-5sin5x)cos3x+(-3sin3x)cos5x =-5sin5xcos3x-3sin3xcos5x

sin5xcos3x-cos5xsin3x=sin(5x-3x)=0 sin2x=0 那么,所有sin值为零的角度为180度的整数倍,所以 2x=k*pi(k属于整数) x=k*pi/2(k属于整数) (pi为圆周率)

∫(sin5x -cos3x)dx =∫sin5x dx -∫cos3xdx =1/5 *∫sin5x d(5x) - 1/3*∫cos3xd(3x) = -1/5 *cos5x -1/3 *sin3x +C,C为常数

使用积化和差公式一步到位 sinxcosy=(sin(x+y)+sin(x-y))/2

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