We need to find the value of Int [(z+1) / (3z^2 + 6z +5) ^
(1/3) dz]
First we substitute 3z^2 + 6z +5 =
u
=> du/dz = 6z +6 =
6*(z+1)
=> du = 6*(z+1)
dz
Therefore we can substitute [(z+1) / (3z^2 + 6z +5) ^
(1/3) dz] with (1/6)/ u^ (1/3) du
=> Int [(z+1) /
(3z^2 + 6z +5) ^ (1/3) dz]
=> Int [(1/6) / u^ (1/3)
du]
=> (1/6) Int [u^ (-1/3)
du]
=> (1/6) [u^ (-1/3 +1) / (-1/3 +1)]
+C
=> (1/6) *u^ (2/3)/ (2/3)
+C
=> (1/6)*(3/2) u^ (2/3)
+C
=> (1/4) u^ (2/3)
+C
=> (1/4) (3z^2 + 6z +5) ^ (2/3)
+C
Therefore the required result is (1/4)
(3z^2 + 6z +5) ^ (2/3) +C
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