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Answer :
The evaluated integral is ∫(5x + 1)/(x² + 5x + 14) dx = A ln|x - râ‚| + B ln|x - râ‚‚| + C. To integrate the partial fractions, we assign variables A and B to the numerator constants.
To evaluate the integral I = ∫(5x + 1)/(x² + 5x + 14) dx, we first need to factor the denominator. However, the quadratic x² + 5x + 14 cannot be factored further using real numbers. Therefore, we proceed with partial fraction decomposition.
We assign variables A and B to the numerator constants and write the partial fraction decomposition as:
(5x + 1)/(x² + 5x + 14) = A/(x - râ‚) + B/(x - râ‚‚)
To determine the values of A and B, we equate the numerators:
5x + 1 = A(x - râ‚‚) + B(x - râ‚)
Expanding and collecting like terms, we obtain:
5x + 1 = (A + B)x - (Arâ‚‚ + Brâ‚) + (Arâ‚râ‚‚)
By equating the coefficients of like powers of x, we get a system of equations:
A + B = 5
-Arâ‚‚ - Brâ‚ = 1
Solving this system of equations, we find the values of A and B.
Once we have the values of A and B, we can rewrite the integral in terms of the partial fractions:
∫(5x + 1)/(x² + 5x + 14) dx = ∫[A/(x - râ‚) + B/(x - râ‚‚)] dx
Integrating each term separately, we obtain:
= A ln|x - râ‚| + B ln|x - râ‚‚| + C
Where C is the constant of integration.
In conclusion, the evaluated integral is ∫(5x + 1)/(x² + 5x + 14) dx = A ln|x - râ‚| + B ln|x - râ‚‚| + C, where A and B are determined through partial fraction decomposition and râ‚ and râ‚‚ are the roots of the quadratic denominator.
To learn more about partial fraction decomposition click here : brainly.com/question/30401234
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