∫t+2tdtSubstitute u=t+2⟶dtdu=2t1⟶dt=2tdu, use:=t=(u−2)22∫u(u−2)2duNow solving:=∫u(u−2)2du∫(u+u4−4)duApply linearity:=∫udu+4∫u1du−4∫1duNow solving:∫uduApply power rule:∫undu=n+1un+1 with n=1=2u2
Now solving:∫u1duThis is a standard integral:=ln(u)Now solving:∫1duApply constant rule:=uPlug in solved integrals:∫udu+4∫u1du−4∫1du=4ln(u)+2u2−4uPlug in solved integrals:2∫u(u−2)2du=8ln(u)+u2−8u Undo substitution u=t+2=−8(t+2)+(t+2)2+8ln(t+2)The problem is solved:∫t+2tdt=−8(t+2)+(t+2)2+8ln(t+2)+C=t−4t+8ln(t+2)+C
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