3)
∫−11(x+21)(x2+x+1)dx
Solution
∫(x+21)(x2+x+1)dx
Substitute. u=x2+x+1
dxdu=2x+1→dx=2x+11du:
=21∫udu
Now solving:
=∫udu
Apply power rule
=∫undu=n+1un+1 with n=21
=32u23
Plug in solved integrals
=21∫udu
=3u23
Undo substitution u=x2+x+1:
=3(x2+x+1)23+C
Inserting the bounds
=[3(12+1+1)23+C]−[3((−1)2+(−1)+1)23+C]
3−31=1.398717474235544
4)
∫(x+3)(x+2)dx
Solution
∫x+3x+2dx
Substitute u=x+3
dxdu=1→dx=du
=∫uu−1du
Expand
=∫[u−u1]du
Apply linearity
=∫udu−∫u1du
Now solving
=∫udu
Apply power rule:
=∫undu=n+1un+1 with n=21
=32u23
Now solving
∫u1du
Apply power rule with n=−21:
=2u
Plug in solved integrals
=∫udu−∫u1du
=32u23−2u
Undo substitution u=x+3:
=32(x+3)23−2x+3
=32(x+3)23−2x+3+C
=2[32(x+3)23−x+3]+C
=32x(x+3)+C
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