# How do you integrate #(x^3+25)/(x^2+4x+3)# using partial fractions?

Now I decomposed

After equating coefficients, I found

After solving them simultaneously,

Thus,

- I took long division

- I decomposed second integral into basic fractions

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To integrate the function ((x^3+25)/(x^2+4x+3)) using partial fractions, follow these steps:

- Factor the denominator (x^2 + 4x + 3) into linear factors: ((x+1)(x+3)).
- Decompose the fraction into partial fractions. Let ((x^3+25)/(x^2+4x+3)) = (\frac{A}{x+1}) + (\frac{B}{x+3}).
- Clear the denominators by multiplying both sides of the equation by ((x+1)(x+3)).
- After clearing the denominators, you'll get the equation (x^3 + 25 = A(x+3) + B(x+1)).
- Expand and simplify the equation to solve for (A) and (B).
- Once you have found the values of (A) and (B), substitute them back into the partial fraction decomposition.
- Now, integrate each term separately.
- Finally, combine the integrals to get the result.

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