Question

求下列不定积分∫x^4/1+x^2dx ∫(2sinx-1/2cosx)dx ∫(1+cos^2x/1+cos2x)dx

Answer 1

∫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