# How do you integrate #x/(1+x^4)#?

Method

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To integrate ( \frac{x}{1+x^4} ), you can use partial fraction decomposition combined with trigonometric substitution.

First, perform partial fraction decomposition to express ( \frac{x}{1+x^4} ) as a sum of simpler fractions.

[ \frac{x}{1+x^4} = \frac{Ax + B}{1+x^2} + \frac{Cx + D}{1+x^2} ]

Then, solve for the constants ( A ), ( B ), ( C ), and ( D ).

After finding the constants, you can substitute ( 1+x^2 = u ) and then use trigonometric substitution to integrate the expression.

Finally, substitute back the value of ( u ) to find the final result of the integral.

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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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