3.6 Q-29

Question Statement

Evaluate the integral:

$$\int_{\pi / 6}^{\pi / 2} \frac{\operatorname{Cos} x}{\operatorname{Sin} x (2 + \operatorname{Sin} x)} , dx$$

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Background and Explanation

To solve this integral, we need to understand the basic trigonometric identities and properties. Specifically:

• Cosecant ($\operatorname{Cosec} x = \frac{1}{\sin x}$) and cotangent ($\operatorname{Cot} x = \frac{\cos x}{\sin x}$) will help simplify the expression.
• The integral involves a rational trigonometric expression, which requires manipulating and simplifying using these identities.
• This type of integral can often be simplified by substitution or recognizing standard integral forms.

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Solution

Step-by-step explanation:

1. Rewrite the integrand:
   We start by rewriting the integrand using trigonometric identities.

$$\frac{\operatorname{Cos} x}{\operatorname{Sin} x (2 + \operatorname{Sin} x)} = \left( \frac{2}{\sin x} + 1 \right)^{-1} (-2 \operatorname{Cosec} x \operatorname{Cot} x)$$

2. Substitute:
   Next, we simplify the expression by noting that $\operatorname{Cosec} x = \frac{1}{\sin x}$ and $\operatorname{Cot} x = \frac{\cos x}{\sin x}$. This leads to:

$$\int_{\pi / 6}^{\pi / 2} (2 \operatorname{Cosec} x + 1)^{-1} (-2 \operatorname{Cosec} x \operatorname{Cot} x) , dx$$

3. Factor and simplify:
   We can simplify the expression further, and focus on the integration:

$$= -\frac{1}{6} \int_{\pi / 6}^{\pi / 2} (2 \operatorname{Cosec} x + 1)^{-1} (-2 \operatorname{Cosec} x \operatorname{Cot} x) , dx$$

4. Apply integration rules:
   Using standard integral rules for logarithmic and trigonometric functions, we get:

$$= -\frac{1}{2} \ln \left( (2 \operatorname{Cosec} x + 1) \right) \bigg|_{\pi / 6}^{\pi / 2}$$

5. Evaluate the bounds:
   Finally, evaluate the logarithmic function at the limits $x = \pi / 2$ and $x = \pi / 6$:

$$= -\frac{1}{2} \left[ \ln \left( \frac{2}{\sin (\pi / 2)} + 1 \right) - \ln \left( \frac{2}{\sin (\pi / 6)} + 1 \right) \right]$$

Simplifying further:

$$= -\frac{1}{2} \left[ \ln (2 + 1) - \ln (4 + 1) \right]$$

6. Final answer:
   This results in:

$$= \frac{1}{2} \left[ \ln 3 - \ln 5 \right] = \frac{1}{2} \ln \frac{5}{3}$$

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Key Formulas or Methods Used

• Trigonometric Identities:
  • $\operatorname{Cosec} x = \frac{1}{\sin x}$
  • $\operatorname{Cot} x = \frac{\cos x}{\sin x}$
• Integral of Logarithms:
  • The integral of the form $\int \frac{dx}{x} = \ln |x|$

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Summary of Steps

1. Rewrite the integrand using trigonometric identities.
2. Simplify the integrand and apply the integral formula for logarithms.
3. Evaluate the integral at the limits $x = \pi / 6$ and $x = \pi / 2$.
4. Simplify the result to obtain $\frac{1}{2} \ln \frac{5}{3}$.