# How do you integrate #int (x^2*2x*2)^2 dx# ?

Let's first make the integrand simpler.

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To integrate ( \int (x^2 \cdot 2x \cdot 2)^2 , dx ), you can simplify the expression inside the integral first:

[ (x^2 \cdot 2x \cdot 2)^2 = (4x^3)^2 = 16x^6 ]

So, the integral becomes:

[ \int 16x^6 , dx ]

To integrate ( 16x^6 ) with respect to ( x ), you can use the power rule for integration:

[ \int x^n , dx = \frac{x^{n+1}}{n+1} + C ]

Where ( n ) is any real number except for ( -1 ).

Applying this rule to ( 16x^6 ), we have:

[ \int 16x^6 , dx = \frac{16x^{6+1}}{6+1} + C = \frac{16x^7}{7} + C ]

So, the integral of ( \int (x^2 \cdot 2x \cdot 2)^2 , dx ) simplifies to ( \frac{16x^7}{7} + 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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