Question

The value of the definite integral $\int_{0}^{\pi} \frac{1}{4 + 3 \cos x} \, dx$ is

• A. $1$
• B. $\frac{\pi}{7}$
• C. $\frac{\pi}{\sqrt{7}}$
• D. $\frac{2}{\sqrt{7}}$

Hint:

For integrals of the type $\int_{0}^{\pi} \frac{1}{a + b \cos x} \, dx$ where $a > b > 0$, you can use the direct formula shortcut: $\text{Result} = \frac{\pi}{\sqrt{a^2 - b^2}}$. Plugging in $a=4$ and $b=3$ gives $\frac{\pi}{\sqrt{16-9}} = \frac{\pi}{\sqrt{7}}$ in one second flat!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question asks us to evaluate the definite integral $I = \int_{0}^{\pi} \frac{1}{4 + 3 \cos x} \, dx$.

Step 2: Key Formula or Approach:
For rational trigonometric integrals of this type involving $\cos x$, we use the standard Weierstrass substitution:
$$\tan\left(\frac{x}{2}\right) = t \implies \cos x = \frac{1-t^2}{1+t^2} \quad \text{and} \quad dx = \frac{2\,dt}{1+t^2}$$ We must also change the integration limits accordingly:
When $x = 0 \implies t = \tan(0) = 0$
When $x = \pi \implies t = \tan\left(\frac{\pi}{2}\right) \rightarrow \infty$

Step 3: Detailed Explanation:
Substitute the algebraic expressions into the integral:
$$I = \int_{0}^{\infty} \frac{1}{4 + 3\left(\frac{1-t^2}{1+t^2}\right)} \cdot \frac{2\,dt}{1+t^2}$$ Take the common denominator in the lower expression:
$$I = \int_{0}^{\infty} \frac{1}{\frac{4(1+t^2) + 3(1-t^2)}{1+t^2}} \cdot \frac{2\,dt}{1+t^2}$$ The $(1+t^2)$ terms in the denominators cancel out perfectly:
$$I = \int_{0}^{\infty} \frac{2}{4 + 4t^2 + 3 - 3t^2} \, dt$$ $$I = \int_{0}^{\infty} \frac{2}{t^2 + 7} \, dt = 2 \int_{0}^{\infty} \frac{1}{t^2 + (\sqrt{7})^2} \, dt$$ Apply the standard integral form $\int \frac{1}{t^2 + a^2} \, dt = \frac{1}{a} \tan^{-1}\left(\frac{t}{a}\right)$:
$$I = 2 \cdot \left[ \frac{1}{\sqrt{7}} \tan^{-1}\left(\frac{t}{\sqrt{7}}\right) \right]_{0}^{\infty}$$ $$I = \frac{2}{\sqrt{7}} \left[ \tan^{-1}(\infty) - \tan^{-1}(0) \right]$$ Since $\tan^{-1}(\infty) = \frac{\pi}{2}$ and $\tan^{-1}(0) = 0$:
$$I = \frac{2}{\sqrt{7}} \cdot \left( \frac{\pi}{2} - 0 \right) = \frac{\pi}{\sqrt{7}}$$ This matches option (C).

Step 4: Final Answer:
The value of the integral is $\frac{\pi}{\sqrt{7}}$, which corresponds to option (C).