∫x4+9x(x2+4)dx Substitute x2=u,2xdx=du
∫x4+9x(x2+4)dx=21∫u2+9u+4du
=21∫u2+9udu+2∫u2+9du
21∫u2+9udu
t=u2+9,dt=2udu
21∫u2+9udu=41∫tdt=
=41ln∣t∣+C1=41ln(u2+9)+C1
=41ln(x4+9)+C1
2∫u2+9du
s=3u,ds=31du
2∫u2+9du=2∫9s2+93ds=32∫1+s2ds
=32tan−1(s)+C2=32tan−1(3u)+C2
=32tan−1(3x2)+C2 Then
∫x4+9x(x2+4)dx=41ln(x4+9)+32tan−1(3x2)+C
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