# How do you integrate #int x/((x^2+1)^2)# using partial fractions?

Rather than using partial fractions, we can make a simple substitution to find that

Rather than partial fractions, this is easiest to solve through integration by substitution.

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To integrate ( \frac{x}{(x^2 + 1)^2} ) using partial fractions, follow these steps:

- Express the given fraction as a sum of simpler fractions using partial fractions.
- Set up the equation ( \frac{x}{(x^2 + 1)^2} = \frac{A}{x^2 + 1} + \frac{Bx + C}{(x^2 + 1)^2} ).
- Multiply both sides of the equation by ( (x^2 + 1)^2 ) to clear the denominators.
- Expand and equate coefficients to solve for ( A ), ( B ), and ( C ).
- Integrate each term separately.
- Finally, combine the results to find the integral of the original function.

Following these steps, we find that:

( A = \frac{1}{2} ), ( B = 0 ), and ( C = -\frac{1}{2} ).

Therefore, the partial fraction decomposition is ( \frac{x}{(x^2 + 1)^2} = \frac{1}{2(x^2 + 1)} - \frac{x/2}{(x^2 + 1)^2} ).

Now, integrate each term separately:

( \int \frac{1}{2(x^2 + 1)} , dx = \frac{1}{2} \tan^{-1}(x) + C_1 ).

( \int -\frac{x/2}{(x^2 + 1)^2} , dx = \frac{1}{4(x^2 + 1)} + C_2 ).

Finally, combine the results:

( \int \frac{x}{(x^2 + 1)^2} , dx = \frac{1}{2} \tan^{-1}(x) - \frac{1}{4(x^2 + 1)} + C ), where ( C = C_1 + C_2 ).

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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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