求下列不定积分∫x^4/1+x^2dx ∫(2sinx-1/2cosx)dx ∫(1+cos^2x/1+cos2x)dx
1个回答
∫x^4/1+x^2dx
=∫[(x^4-1)/(1+x^2）+1/(1+x^2)]dx
=∫(x^2-1)dx+∫1/(1+x^2)dx
=x^3/3-x+arctanx+C
∫(2sinx-1/2cosx)dx
=∫tanxdx-1/2∫secxdx
=-ln|cosx|-1/2ln|secx+tanx|+C
∫(1+cos^2x/1+cos2x)dx
=∫(1+cos^2x)/(2cosx^2)dx
=1/2∫sec^2xdx+∫(1/2)dx
=(tanx)/2 + x/2 + C
∫(cos2x/sin^2x*cos^2x)dx
=∫ (cos^2x-sin^2x)/(sin^2x*cos^2x)dx
=∫csc^2xdx-∫sec^2xdx
=-cotx-tanx + C
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