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∫xCos^3 xDx

∫xcos^3 xdx =∫xcos^2 xcosxdx =∫xcos^2 xdsinx =∫x(1-sin^2 x)dsinx =∫xdsinx-∫xsin^2 x dsinx =xsinx-∫sinxdx-1/3∫xdsin^3 x =xsinx+∫dcosx-1/3(xsin^3 x -∫sin^3 x dx) =xsinx+cosx-xsin^3 x/3+1/3∫sin^2xsinxdx =xsinx+cosx-xsin^3 x/3-1/3∫...

用两次分步积分法 ∫x^2cos3xdx =1/3∫x^2dsin3x =1/3x^2sin3c-2/3∫xsin3xdx =1/3x^2sin3c+2/3∫xdcos3x =1/3x^2sin3c+2/3xcos3x-2/3∫cos3xdx =1/3x^2sin3c+2/3xcos3x-2/9sin3x+C

第一类换元法就是凑微分法 ∫sinxdx/cos³x =-∫d(cosx)/cos³x =(1/2)∫d(1/cos²x) =(1/2)*(1/cos²x)+C =1/(2cos²x)+C

∫(1-sinx^2)d(sinx)=sinx-1/3sinx^3

记A=∫e^2xsin3xdx 用分部积分法: A=0.5e^(2x)sin3x-∫0.5e^(2x)3cos3xdx =0.5e^(2x)sin3x-1.5∫e^(2x)cos3xdx =0.5e^(2x)sin3x-1.5[0.5e^(2x)cos3x+∫0.5e^(2x)3sin3xdx] =0.5e^(2x)sin3x-0.75e^(2x)cos3x-2.25∫e^(2x)sin3xdx =0.5e^(2x)sin3x-0.7...

d(cosx^3)=3*cosx^2*-sinx*dx,不等于sinx^2

x^2*sin3xdx = -1/3∫x^2dcos3x = -1/3x^2cos3x+2/3∫xcos3xdx = -1/3x^2cos3x+2/9∫xdsin3x = -1/3x^2cos3x+2/9xsin3x-2/9∫sin3xdx = -1/3x^2cos3x+2/9xsin3x+2/27cos3x+C

∫(sinx)^3.(cosx)^5dx = ∫-(1-(cosx)^2) (cosx)^5 dcosx = (cosx)^8/8 - (cosx)^6/6 +C

你好 令√x=t,则x=t²,dx=2tdt ∫cos√xdx=2∫tcostdt=2∫td(sint)=2[tsint-∫sintdt]=2tsint+2cost+C,即原式=2√xsin√x+2cos√x+C

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