Question

Evaluate the integral: \[ \int \frac{dx}{4 + 5 \cos x} \]

• A. \( -\frac{1}{3} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \)
• B. \( \frac{1}{3} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \)
• C. \( -\frac{1}{9} \log \left| \frac{3 - \tan \frac{x}{2}}{3 + \tan \frac{x}{2}} \right| + C \)
• D. \( -\frac{1}{9} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \)

Hint:

For integrals involving trigonometric functions, try substituting \( t = \tan \frac{x}{2} \) to simplify the expression and use standard results for the integral.

Answer 1

The Correct Option is B

The given integral is \( \int \frac{dx}{4 + 5 \cos x} \). Step 1: Use the substitution \( t = \tan \frac{x}{2} \), which simplifies the trigonometric expression. Step 2: The integral becomes: \[ \int \frac{dx}{4 + 5 \cos x} = \frac{1}{3} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \] % Final Answer The correct integral is \( \frac{1}{3} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \).

Answer 2

Step 1: Use Weierstrass substitution
Let \( t = \tan \frac{x}{2} \), then:
\[ \cos x = \frac{1 - t^2}{1 + t^2}, \quad dx = \frac{2}{1 + t^2} dt \]

Step 2: Rewrite the integral in terms of \( t \)
\[ \int \frac{dx}{4 + 5 \cos x} = \int \frac{\frac{2}{1 + t^2} dt}{4 + 5 \cdot \frac{1 - t^2}{1 + t^2}} = \int \frac{2 \, dt}{(1 + t^2) \left(4 + 5 \frac{1 - t^2}{1 + t^2}\right)} \]
Simplify denominator:
\[ 4 + 5 \frac{1 - t^2}{1 + t^2} = \frac{4(1 + t^2) + 5(1 - t^2)}{1 + t^2} = \frac{4 + 4t^2 + 5 - 5 t^2}{1 + t^2} = \frac{9 - t^2}{1 + t^2} \]
So integral becomes:
\[ \int \frac{2 dt}{(1 + t^2) \cdot \frac{9 - t^2}{1 + t^2}} = \int \frac{2 dt}{9 - t^2} \]

Step 3: Integrate the simplified integral
\[ \int \frac{2}{9 - t^2} dt = \int \frac{2}{(3)^2 - t^2} dt \] This is a standard integral:
\[ \int \frac{dt}{a^2 - t^2} = \frac{1}{2a} \log \left| \frac{a + t}{a - t} \right| + C \] So:
\[ \int \frac{2}{9 - t^2} dt = \frac{2}{2 \cdot 3} \log \left| \frac{3 + t}{3 - t} \right| + C = \frac{1}{3} \log \left| \frac{3 + t}{3 - t} \right| + C \]

Step 4: Substitute back \( t = \tan \frac{x}{2} \)
\[ \int \frac{dx}{4 + 5 \cos x} = \frac{1}{3} \log \left| \frac{3 + \tan \frac{x}{2}}{3 - \tan \frac{x}{2}} \right| + C \]