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Solve the following.
Evaluate: ∫ (2x + 7)/(x² - x - 2) dx.
Evaluate: $\int \frac{2x + 7}{x^2 - x - 2} \, dx$.
Answer
Answer
Integral: $\int \left((2x + 7)/(x^2 - x - 2)\right) dx$
$x^2 - x - 2 = (x - 2)(x + 1)$. Partial fractions:
$\frac{2x + 7}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1}$, so $2x + 7 = A(x + 1) + B(x - 2)$.
Put $x = 2$: $11 = 3A$, so $A = \frac{11}{3}$. Put $x = -1$: $5 = -3B$, so $B = -\frac{5}{3}$.
$\int\frac{2x + 7}{x^2 - x - 2}dx = \frac{11}{3}\int\frac{dx}{x - 2} - \frac{5}{3}\int\frac{dx}{x + 1}$
Hence the integral is $\frac{11}{3}\ln|x - 2| - \frac{5}{3}\ln|x + 1| + C$.
Final answer: $\frac{11}{3}\ln|x - 2| - \frac{5}{3}\ln|x + 1| + C$