The indefiniteintegral ∫y24-y32/3dy
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
Here, use the substitution method because the differential of the function 4-y3 is -3y2dy and y2dy is present in the integral.
Differentiate u=4-y3 with respect to y.
As y2 dy is a part of the integration, solving for y2 dy by dividing both side by -3.
By using concept i),
substitute u=4-y2, y2dy=-du3 to get
By using concept ii),
Cancelling out common factor: