Question

Evaluate the indefinite integral. (Use C for the constant of integration.)?

Answer 1

The integral equals `3/8(z^4 + 7)^(2/3) + C`

Explanation:

We let `u = z^4 + 7`. Then `du = 4z^3dz -> dz = (du)/(4z^3)`.

`I = int z^3/root(3)(u) * 1/(4z^3) du`

`I = 1/4 int 1/root(3)(u)du`

`I = 1/4int u^(-1/3)`

By the power rule of integration:

`I = 1/4(3/2)u^(2/3) + C`

Reverse the substitution.

`I = 3/8(z^4+ 7)^(2/3) + C`

Hopefully this helps!