How do you integrate #intx^2/(1+x^3)^2# using substitution?
Integral becomes
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To integrate ( \frac{x^2}{(1+x^3)^2} ) using substitution, follow these steps:

Let ( u = 1 + x^3 ). Then, ( du = 3x^2 , dx ).

Solve for ( dx ): ( dx = \frac{du}{3x^2} ).

Rewrite the integral in terms of ( u ): ( \int \frac{x^2}{(1+x^3)^2} , dx = \int \frac{x^2}{u^2} \cdot \frac{du}{3x^2} ).

Simplify: ( \frac{x^2}{x^2} = 1 ).

Cancel out ( x^2 ) terms and rewrite the integral in terms of ( u ): ( \int \frac{1}{u^2} , du ).

Integrate: ( \int \frac{1}{u^2} , du = \frac{1}{u} + C ).

Substitute back for ( u ): ( \frac{1}{1+x^3} + C ).
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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