# How do you integrate #int(4x)/(x^4 +1) # using partial fractions?

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To integrate ( \int \frac{4x}{x^4 + 1} ) using partial fractions, first factor the denominator:

( x^4 + 1 = (x^2 + 1)^2 - 2x^2 = (x^2 - \sqrt{2}x + 1)(x^2 + \sqrt{2}x + 1) )

Now, we can express ( \frac{4x}{x^4 + 1} ) as partial fractions:

( \frac{4x}{x^4 + 1} = \frac{Ax + B}{x^2 - \sqrt{2}x + 1} + \frac{Cx + D}{x^2 + \sqrt{2}x + 1} )

By comparing numerators, we get:

( 4x = (Ax + B)(x^2 + \sqrt{2}x + 1) + (Cx + D)(x^2 - \sqrt{2}x + 1) )

Solve for ( A ), ( B ), ( C ), and ( D ) using algebraic manipulation.

Once you find the values of ( A ), ( B ), ( C ), and ( D ), you can integrate each term separately. The integrals will likely involve natural logarithms and arctangent functions.

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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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