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Evaluate the integral.
∫d x/x²-4 x+9 (6) ∫x³-3 x²+2 x-3/x²+1 d x

1 Answer

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Final answer:

To evaluate the integral of a rational function involving a quadratic term in the denominator, we can use partial fraction decomposition and then integrate each term separately.

Step-by-step explanation:

To evaluate the integral ∫(x³-3x²+2x-3)/(x²+1) dx, we can use partial fraction decomposition and then integrate each term separately. First, factor the denominator x²+1 as (x+i)(x-i), where i is the imaginary unit. Then, we can write the given expression as (A(x+i) + B(x-i))/(x²+1), where A and B are constants. We can find the values of A and B by equating the numerators of the decomposed expression and comparing the coefficients of x. After finding A and B, we can integrate each term separately and obtain the final result.

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