Question

Evaluate the following integrals: $$ \int \frac{\cos x d x}{\sin ^{2} x-6 \sin x+12} d x $$

Short answer

Question: Use trigonometric identities and substitution to simplify the integrand of the following integral and describe the steps involved in evaluating it: \(\int \frac{\cos x}{\sin^2(x) - 6\sin(x) + 12} dx\) Answer: The integrand is simplified to \(\frac{\cos x}{1 - \cos^2(x) - 6\sin x + 12}\). After performing a substitution, the integral becomes \(\int \frac{-du}{1 - u^2 - 6\sqrt{1 - u^2} + 12}\). Evaluating this integral analytically is challenging, and numerical methods such as Simpson's rule or the trapezoidal rule can be used to approximate its value.

Step-by-step solution

Apply the double angle formula for sine to the denominator of the integrand

Since \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\), we can write \(\sin^2(x) - 6\sin(x) + 12\) as $$ \frac{1}{4}\left[(2\sin(x) - 6)^2 - 6^2 + 48\right]. $$

Simplify the denominator using the binomial formula

Using the binomial formula, we can simplify \((2\sin x - 6)^2\) and obtain $$ (2\sin x - 6)^2 = 4\sin^2x - 24\sin x + 36. $$ So the denominator simplifies to $$ \frac{1}{4}\left[4\sin^2x - 24\sin x + 36 - 36 + 48\right] = \sin^2(x) - 6\sin(x) + 12. $$

Apply the Pythagorean identity and rewrite the integrand

Using the identity \(\sin^2(\theta) = 1 - \cos^2(\theta)\), we can rewrite the denominator as $$ \sin^2(x) - 6\sin(x) + 12 = (1 - \cos^2(x)) - 6\sin x + 12. $$ Then the integrand becomes $$ \frac{\cos x}{1 - \cos^2(x) - 6\sin x + 12}. $$

Perform a substitution

Let \(u = \cos x\), then \(-du = \sin x\,dx\). The integral becomes $$ \int \frac{-du}{1 - u^2 - 6\sqrt{1 - u^2} + 12}. $$

Evaluate the integral

The integral is now a rational function with a term containing a square root. Since the function is not easily integrable, and we have exhausted our trigonometric identities and substitution options, we can apply numerical methods, such as Simpson's rule or the trapezoidal rule, to approximate the integral. Note that such methods are beyond the scope of this high school course. To summarize, the given integral is challenging to evaluate analytically, and numerical methods are recommended for obtaining an approximate value.