Question

Dx 54.1) 2) 3) cos(-a) · sin ( 1 ) dx; 4

Answer 1

1.∫(4-3cosx)dx=4x-3sinx+C

2.∫cos4xdx=1/4∫cosudu=sinu/4+C=1/4*sin(4x)+C

u=4x

du=4dx

3.∫cos(x/6)dx=6∫cosudu=6sinu+C=6sin(x/6)+C

u=x/6

du=1/6*dx

4.∫cos(2-3x)dx=-1/3∫cosudu=-sinu/3+C=-1/3*sin(2-3x)+C

u=2-3x

du=-3dx

5.∫x²cosx³dx=1/2∫x²cosx(cos2x+1)dx=1/2∫(x²cosx+x²cosxcos2x)dx=1/2∫x²cosxcos2xdx+1/2∫x²cosxdx=1/4∫x²(cosx+cos3x)dx+1/2∫x²cosxdx=1/4∫(x²cosx+x²cos3x)dx+1/2∫x²cosxdx=1/4∫x²cos3xdx+3/4∫x²cosxdx=1/12*x²*sin3x-1/6∫xsin3xdx+3/4∫x²cosxdx=1/18*x*cos3x+1/12*x²*sin3x-1/18∫cos3xdx+3/4∫x²cosxdx=1/18*x*cos3x+1/12*x²*sin3x-1/54∫cosudu+3/4∫x²cosxdx=1/18*x*cos3x+1/12*x²*sin3x-sinu/54+3/4∫x²cosxdx=1/18*x*cos3x+1/12*x²*sin3x-sinu/54+3/4*x²sinx-3/2∫xsinxdx=1/18*x*cos3x+1/12*x²*sin3x-sinu/54+3/4*x²sinx+3/2xcosx-3/2∫cosxdx=1/18*x*cos3x+1/12*x²*sin3x-sinu/54+3/4*x²sinx+3/2*xcosx-3sinx/2+C=3/4*x²sinx+1/12*x²*sin3x-3sinx/2-1/54*sin3x+3/2*x*cosx+1/18*x*cos3x+C

u=3x

du=3dx