# How do you integrate #(x^2 + 2x -5) / [(x - 2)*(x+1)*(x^2+1)]#?

I am not 100% sure for this one but I have attached a picture so that you may see what I did.

Please, feel free to ask any questions and even more, to correct what I did wrong :)

Integrals can be a pain lol

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To integrate ((x^2 + 2x -5) / [(x - 2)(x+1)(x^2+1)]), you first need to perform partial fraction decomposition to express the rational function as a sum of simpler fractions. Then, you can integrate each term separately. The partial fraction decomposition of the given expression would be:

[\frac{x^2 + 2x -5}{(x - 2)(x+1)(x^2+1)} = \frac{A}{x - 2} + \frac{B}{x + 1} + \frac{Cx + D}{x^2 + 1}]

Solving for (A), (B), (C), and (D) by equating coefficients, you'll find:

(A = \frac{3}{10}), (B = -\frac{3}{10}), (C = \frac{1}{2}), (D = -\frac{1}{2})

So, the integral becomes:

[\int \frac{3}{10(x - 2)} - \frac{3}{10(x + 1)} + \frac{\frac{1}{2}x - \frac{1}{2}}{x^2 + 1} ,dx]

Now you can integrate each term separately:

[\frac{3}{10}\ln|x - 2| - \frac{3}{10}\ln|x + 1| + \frac{1}{4}\ln(x^2 + 1) + 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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