3.3 Q-21

Question Statement

Evaluate the integral:

$$\int \frac{\sqrt{3}}{\sin x + \cos x} , dx$$

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Background and Explanation

To solve this integral, we need to simplify the expression in the denominator. Using trigonometric identities and appropriate substitutions will make the problem more manageable. The goal is to express the denominator in a way that will allow us to integrate using known standard integrals. Specifically, we’ll use a well-known identity to simplify the terms and make use of the secant function.

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Solution

Let’s break the solution down step by step:

1. Factor Out Constants: First, notice that we can factor out the constants in the numerator and denominator to simplify the expression:

$$\int \frac{\sqrt{3}}{\sin x + \cos x} , dx = \frac{1}{\sqrt{2}} \int \frac{\sqrt{2}}{\frac{\sin x + \cos x}{\sqrt{2}}} , dx$$

2. Use Trigonometric Identity: The expression inside the denominator, $\frac{\sin x + \cos x}{\sqrt{2}}$, can be rewritten using a standard trigonometric identity. Recall the identity for $\sin(x + \frac{\pi}{4})$, which states:

$$\sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4} = \sin(x + \frac{\pi}{4})$$

Applying this identity, we rewrite the integral as:

$$\int \frac{1}{\sin x \sin \frac{\pi}{4} + \cos x \cos \frac{\pi}{4}} , dx = \int \frac{1}{\cos \left(x - \frac{\pi}{4}\right)} , dx$$

3. Apply the Secant Identity: Now, the integrand is in a familiar form. Using the identity $\sec \theta = \frac{1}{\cos \theta}$, we get:

$$\int \sec \left(x - \frac{\pi}{4}\right) , dx$$

4. Integrate Using the Standard Integral: The integral of $\sec(x)$ is a standard result:

$$\int \sec x , dx = \ln \left| \sec x + \tan x \right| + C$$

Applying this formula to our integral, we obtain:

$$\ln \sec \left(x - \frac{\pi}{4}\right) + \tan \left(x - \frac{\pi}{4}\right) + C$$

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Key Formulas or Methods Used

• Trigonometric Identity:
  • $\sin(x + \frac{\pi}{4}) = \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4}$
• Secant Function:
  • $\sec(x) = \frac{1}{\cos(x)}$
• Standard Integral:
  • $\int \sec x , dx = \ln |\sec x + \tan x| + C$

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Summary of Steps

1. Factor out constants to simplify the expression.
2. Use the trigonometric identity to rewrite the denominator.
3. Rewrite the integral as $\int \sec(x - \frac{\pi}{4}) , dx$.
4. Apply the standard integral for $\sec x$.
5. The final result is:

$$\ln \sec \left(x - \frac{\pi}{4}\right) + \tan \left(x - \frac{\pi}{4}\right) + C$$