# How do you integrate #8x(x²+1)³ dx #?

We use the Method of Substn.

On the first hand, these two Answers may look different, but,

bearing in mind the binomial expansion of

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To integrate ( 8x(x^2 + 1)^3 ) with respect to ( x ), you can use the substitution method. Let ( u = x^2 + 1 ). Then, ( du/dx = 2x ), and rearranging gives ( x = (u - 1)^{1/2} ).

Now, substitute ( u = x^2 + 1 ) and ( du = 2x , dx ) into the integral:

[ \int 8x(x^2 + 1)^3 , dx = 4 \int u^3 , du ]

Integrate ( u^3 ) with respect to ( u ), then resubstitute ( x ) back in:

[ = 4 \left( \frac{u^4}{4} \right) + C ] [ = u^4 + C ] [ = (x^2 + 1)^4 + C ]

So, the result of the integral is ( (x^2 + 1)^4 + C ), where ( C ) is the constant of integration.

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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.

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.

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.

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