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

Answer 1

Cameron Alexander

#int x/(1+x^4)dx =1/2arctan(x^2)#

Method

#int x/(1+x^4)dx = 1/2int (2xdx)/(1+(x^2)^2)=1/2int (d(x^2))/(1+(x^2)^2) ="I"#

let #x^2 =u#

#=>"I"= 1/2int (du)/(1+u^2) = 1/2arctanu = 1/2arctan(x^2)#

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Answer 2

Ezekiel Alexander

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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Answer from HIX Tutor

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.