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Solve the following.
Evaluate: ∫ (x + 7)/(x² + 4x + 7) dx
Evaluate: $\int \frac{x + 7}{x^2 + 4x + 7} dx$
Answer
Answer
Integral: $\int \left((x+7)/(x^2+4x+7)\right) dx$
Write $x + 7 = \frac12(2x+4) + 5$.
$\int\frac{x+7}{x^2+4x+7}dx = \frac12\int\frac{2x+4}{x^2+4x+7}dx + 5\int\frac{dx}{(x+2)^2 + 3}$
$= \frac12\log|x^2+4x+7| + \frac{5}{\sqrt3}\tan^{-1}\frac{x+2}{\sqrt3} + C$
Hence the integral is $\frac12\log(x^2+4x+7) + \frac{5}{\sqrt3}\tan^{-1}\frac{x+2}{\sqrt3} + C$.
Final answer: $\frac12\log(x^2+4x+7) + \frac{5}{\sqrt3}\tan^{-1}\frac{x+2}{\sqrt3} + C$