#### To determine

**To evaluate:**

The indefiniteintegral ∫y24-y32/3dy

#### Answer

**-154-y353+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

**2) Given:**

∫y24-y32/3dy

**3) Calculation:**

Here, use the substitution method because the differential of the function 4-y3 is -3y2dy and y2dy is present in the integral.

Substitute u=4-y3.

Differentiate u=4-y3 with respect to y.

du=-3y2dy

As y2 dy is a part of the integration, solving for y2 dy by dividing both side by -3.

-du3=y2dy

By using concept i),

substitute u=4-y2, y2dy=-du3 to get

∫y24-y32/3dy=∫u23-du3=-13∫u23du

By using concept ii),

=-13u23+123+1+C

=-13u5353+C

Cancelling out common factor:

=-u535+C

Resubstitute u=4-y3.

=-154-y353+C

**Conclusion:**

Therefore,

∫y24-y32/3dy=-154-y353+C