Question

The value of the definite integral \(\int _0^π\frac{1}{5+4cosx}\,dx\) is equal to ...

• A. \(π\)
• B. \(\frac{π}{2}\)
• C. \(\frac{π}{3}\)
• D. \(\frac{π}{4}\)

Hint:

Split at x = 4 where the absolute value changes form.

Answer 1

The Correct Option is C

Step 1: Split the region:
For \(3 \le x \le 4\), \(y = 4 - x\). For \(4 \le x \le 5\), \(y = x - 4\).

Step 2: Integrate:
\[ \int_3^4(4 - x)\,dx = \left[4x - \frac{x^2}{2}\right]_3^4 = (16 - 8) - (12 - 4.5) = \frac12 \]
\[ \int_4^5(x - 4)\,dx = \left[\frac{x^2}{2} - 4x\right]_4^5 = (12.5 - 20) - (8 - 16) = \frac12 \]

Step 3: Total:
Total area = \(\frac12 + \frac12 = 1\) square unit. The shape is two small right triangles, each with legs 1 and 1.

Final Answer:
The area is 1 square unit, option (D). \[ \boxed{1\text{ sq. unit}} \]