Question

Evaluate the integral \( \int \frac{2 - \sin x}{2 \cos x + 3} \, dx \):

• A. \(\frac{2}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{3}} \tan \frac{x}{2}\right) - \log \sqrt{2 \cos x + 3} + C\)
• B. \(\frac{4}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right) + \log \sqrt{2 \cos x + 3} + C\)
• C. \(\frac{3}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right) + \log \sqrt{2} \cos x - 3 + C\)
• D. \(\frac{1}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right) - \log \sqrt{2 \cos x - 3} + C\)

Hint:

When dealing with integrals involving trigonometric functions and square roots, substitution and simplification can transform the integral into a more manageable form. Always look for standard substitutions such as \( u = \tan \frac{x}{2} \) to simplify the expressions.

Answer 1

The Correct Option is B

We are tasked with solving the integral: \[ I = \int \frac{2 - \sin x}{2 \cos x + 3} \, dx. \] Step 1: Substitution Start by using the substitution \( u = \tan \frac{x}{2} \), which transforms the trigonometric functions in terms of \( u \). The relations are: \[ \sin x = \frac{2u}{1+u^2}, \quad \cos x = \frac{1-u^2}{1+u^2}, \quad dx = \frac{2 \, du}{1 + u^2}. \] Step 2: Substituting into the integral Substituting these expressions into the integral: \[ I = \int \frac{2 - \frac{2u}{1+u^2}}{\frac{2(1-u^2)}{1+u^2} + 3} \cdot \frac{2 \, du}{1 + u^2}. \] Step 3: Simplification Simplify the numerator and denominator: \[ {Numerator: } 2 - \frac{2u}{1+u^2} = \frac{2(1+u^2) - 2u}{1+u^2} = \frac{2 + 2u^2 - 2u}{1+u^2}. \] \[ {Denominator: } \frac{2(1-u^2)}{1+u^2} + 3 = \frac{2(1 - u^2) + 3(1 + u^2)}{1 + u^2} = \frac{2 - 2u^2 + 3 + 3u^2}{1 + u^2} = \frac{5 + u^2}{1 + u^2}. \] Thus, the integral becomes: \[ I = \int \frac{\frac{2 + 2u^2 - 2u}{1+u^2}}{\frac{5 + u^2}{1+u^2}} \cdot \frac{2 \, du}{1+u^2} = \int \frac{2 + 2u^2 - 2u}{5 + u^2} \cdot \frac{2 \, du}{1+u^2}. \] Step 4: Perform the integration Now perform the integration. Using standard integration techniques (or integral tables) for such rational functions, we can obtain the final result: \[ I = \frac{4}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right) + \log \left( \sqrt{2 \cos x + 3} \right) + C. \] Thus, the correct answer is: \[ \boxed{\frac{4}{\sqrt{5}} \tan^{-1}\left(\frac{1}{\sqrt{5}} \tan \frac{x}{2}\right) + \log \sqrt{2 \cos x + 3} + C}. \]