# How will you integrate ? #int(dx)/(1+x^4)^2#

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- I divided denominator and numerator of integrand with x^8

- I decomposed numerator for resembling derivative of denominator.

- I used partial fraction

- I expanded fractions with x^4

- I started to decompose second integral by multiply and divide with 2

- I divided denominator and numerator of second integrand with x^2

- I integrated decomposed them.

- I rewrote results.

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To integrate ( \int \frac{dx}{(1+x^4)^2} ), we can use a substitution. Let ( u = x^2 ), then ( du = 2x dx ). This allows us to rewrite the integral as ( \frac{1}{2} \int \frac{du}{(1+u^2)^2} ).

Now, let's use another substitution ( v = \arctan(u) ), then ( dv = \frac{du}{1+u^2} ). This transforms the integral into ( \frac{1}{2} \int dv ), which is simply ( \frac{1}{2} v + C ).

Substitute back for ( u ) and ( v ), we get ( \frac{1}{2} \arctan(u) + C ), where ( u = x^2 ). So, the final result is ( \frac{1}{2} \arctan(x^2) + 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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