Question

consider the integral
\\(\int \sqrt{x^2 - 25} \\, dx\\)
which of the following integration techniques is the best method to simplify the integral?
a. trigonometric substitution with \\(x = 5 \tan \theta\\)
b. trigonometric substitution with \\(x = 5 \sec \theta\\)
c. \\(u\\)-substitution with \\(u = x^2 - 25\\)
d. trigonometric substitution with \\(x = \sin \theta\\)
e. integration by parts with \\(u = \sqrt{x^2 - 25}\\) and \\(dv = \\, dx\\)
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Explanation:

To simplify the integral \(\int \sqrt{x^2 - 25} \, dx\), we analyze the form of the integrand. The expression \(x^2 - a^2\) (here \(a = 5\)) suggests using trigonometric substitution with \(x=a\sec\theta\). For \(x = 5\sec\theta\), we have \(dx=5\sec\theta\tan\theta d\theta\) and \(x^2 - 25=25\sec^2\theta - 25 = 25(\sec^2\theta - 1)=25\tan^2\theta\), so \(\sqrt{x^2 - 25}=5\tan\theta\) (for appropriate \(\theta\)). This substitution simplifies the square - root.

• Option A: \(x = 5\tan\theta\) is used for integrals of the form \(\sqrt{x^2 + a^2}\), not \(\sqrt{x^2 - a^2}\), so A is incorrect.
• Option C: If we use \(u=x^2 - 25\), then \(du = 2xdx\), and there is no \(x\) in the integrand to make this substitution work, so C is incorrect.
• Option D: \(x=\sin\theta\) does not match the form of the integrand \(x^2 - 25\), so D is incorrect.
• Option E: Integration by parts with \(u=\sqrt{x^2 - 25}\) and \(dv = dx\) will lead to a more complicated integral, so E is incorrect.

Answer:

B. Trigonometric substitution with \(x = 5\sec\theta\)