Mcguire Adkins
09/15/2024 · Primary School
\[ I=\int \frac{d x}{\sqrt{x^{2}-4 x+20}} \] \[ f(x)=x^{2}-4 x+20 \] (a) First complete the square of (b) Then use trigonometric substitution to determine the integral. \( \int \sec \theta \mathrm{d} \theta=\ln |\sec \theta+\tan \theta+\mathrm{c}| \)
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To solve the integral \(I = \int \frac{dx}{\sqrt{x^2 - 4x + 20}}\), complete the square to get \(f(x) = (x - 2)^2 + 16\). Use the trigonometric substitution \(x - 2 = 4\tan(\theta)\) to transform the integral into \(\int \sec(\theta) d\theta\), which evaluates to \(\ln |\sec(\theta) + \tan(\theta)| + C\). Back-substitute to get the final result \(I = \ln \left| \sqrt{x^2 - 4x + 20} + x - 2 \right| + C'\).
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