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How do you integrate #int-9x^2(-3x^3+1)^3# using substitution?

How do you integrate #int-9x^2(-3x^3+1)^3dx# using substitution?

Answer 1

Samantha Adkison

#(-3x^3+1)^4/4+C.#

Let us write #I=int-9x^2(-3x^3+1)^3dx.#

Sbstn. #(-3x^3+1)=y rArr -9x^2dx=dy.# Hence,

#I=inty^3dy#

#=y^4/4#

#=(-3x^3+1)^4/4+C.#

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Answer 2

William Abdalla

To integrate ( \int -9x^2(-3x^3+1)^3 ) using substitution, we let ( u = -3x^3 + 1 ). Then, ( du = -9x^2 dx ). Substituting ( u ) and ( du ), the integral becomes ( \int u^3 du ), which is straightforward to integrate. After integrating, we substitute back ( u = -3x^3 + 1 ) to get the final result.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.