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Solve the following.
Evaluate: ∫ (6x+7)/(√(x-5)(x-4)) dx
Evaluate: $\int \frac{6x+7}{\sqrt{(x-5)(x-4)}} \, dx$
Answer
Answer
Integral: $\int \left(\frac{6x+7}{\sqrt{(x-5)(x-4)}}\right) dx$
$x^2 - 9x + 20 = \left(x - \frac{9}{2}\right)^2 - \frac{1}{4}$.
Write $6x + 7 = 3(2x - 9) + 34$.
$I = 3\int\frac{2x-9}{\sqrt{x^2-9x+20}}dx + 34\int\frac{dx}{\sqrt{\left(x-\frac{9}{2}\right)^2 - \left(\frac{1}{2}\right)^2}}$
$= 6\sqrt{x^2-9x+20} + 34\log\left|x - \frac{9}{2} + \sqrt{x^2-9x+20}\right| + C$.
Hence the integral is as above.
Final answer: $6\sqrt{x^2-9x+20} + 34\log\left|x - \frac{9}{2} + \sqrt{x^2-9x+20}\right| + C$