   Chapter 4.5, Problem 14E

Chapter
Section
Textbook Problem

# Evaluate the indefinite integral. ∫ y 2 ( 4 − y 3 ) 2 / 3 d y

To determine

To evaluate:

The indefiniteintegral y24-y32/3dy

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

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