Mathematics

Mathematics

Problem

Evaluate the integral I=∫5x−4x2+2x+2 dx.I=\int \frac{5x-4}{x^{2}+2x+2}\,dx. Answer the parts below in order: (i) Rewrite the quadratic denominator in completed‐square form. (ii) Express the integrand as a sum of two simpler fractions involving constants AA and BB to be determined. (iii) Hence, obtain the exact value of II in terms of elementary logarithmic and inverse‐trigonometric functions, simplifying your final result.

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