To determine

To evaluate:

The indefinite integral ∫z21+z33dz

Answer

121+z323+C

Explanation

1) Concept:

i) The substitution rule

If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then ∫f(gx)g'xdx=∫f(u)du.

ii) Indefinite integral

∫xn dx=xn+1n+1+C  (n≠-1)

2) Given:

∫z21+z33dz

3) Calculation:

Here, use the substitution method because the differential of the function 1+z3 is 3z2dz.

Substitute u=1+z3.

Differentiate u=1+z3 with respect to z.

du=3z2dz

As z2 dz is a part of the integration, solving for z2 dz by dividing both side by 3.

du3=z2dz

By using concept i),

Substitute u=1+z3,  z2dz=du3. to get ∫z21+z33dz=∫1u3du3

Rearranging,

=13∫u-13du

By using concept ii),

=13u-13+1-13+1+C

By simplifying, we get

=13u2323+C

=u232+C

Resubstitute u=1+z3.

=1+z3232+C

= 121+z323+C

Conclusion:

Therefore,

∫z21+z33dz=121+z323+C