To find the values of A, B, C, D, and E, we can multiply both sides of the equation by the common denominator and equate the numerators. Then, we can solve the resulting system of equations to find the values of A, B, C, D, and E.
Once we have the values of A, B, C, D, and E, we can integrate each term separately. The integral of A / (x + 2) is A ln|x + 2|, and the integrals of (Bx + C) / (x + i) and (Dx + E) / (x - i) can be found using logarithmic and inverse tangent functions.
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Explanation:
To integrate the expression (x² + x + 1) / [(x + 2)(x² + 1)], we can use partial fraction decomposition. First, let's factor the denominator:
(x + 2)(x² + 1) = (x + 2)(x + i)(x - i)
Now, let's decompose the fraction into partial fractions:
(x² + x + 1) / [(x + 2)(x² + 1)] = A / (x + 2) + (Bx + C) / (x + i) + (Dx + E) / (x - i)
To find the values of A, B, C, D, and E, we can multiply both sides of the equation by the common denominator and equate the numerators. Then, we can solve the resulting system of equations to find the values of A, B, C, D, and E.
Once we have the values of A, B, C, D, and E, we can integrate each term separately. The integral of A / (x + 2) is A ln|x + 2|, and the integrals of (Bx + C) / (x + i) and (Dx + E) / (x - i) can be found using logarithmic and inverse tangent functions.
I hope this helps! Let me know if you have any further questions.
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