Question

The function \(f(x) = \int _0^x\frac{dt}{1+cost}\) satisfies which of the following differential equations?

• A. \(2\frac{df}{dx} = 1+[f(x)]^2\)
• B. \(\frac{df}{dx} = 1+[f(x)]^2\)
• C. \(2\frac{df}{dx} = 1-[f(x)]^2\)
• D. \(\frac{df}{dx} = 1-[f(x)]^2\)

Hint:

Split the interval where sin 2x changes sign and use symmetry.

Answer 1

The Correct Option is A

Step 1: Use the period:
$|\sin2x|$ has period $\frac\pi2$, so over $[0, \pi]$ there are two identical arches.

Step 2: Area of one arch:
One arch is $\int_0^{\pi/2}\sin2x\,dx = 1$. Two arches give 2 (C).

Final Answer:
2. \[ \boxed{2} \]