Question

The value of $\int_0^{\pi/2} \frac{(\cos x)\sin x}{(\cos x)\sin x + (\sin x)\cos x} \, dx$ is

• A. π/4
• B. 0
• C. π/2
• D. 1/2

Answer 1

The Correct Option is A

We are tasked with evaluating the following integral: \[ I = \int_0^{\pi/2} \frac{(\cos x)\sin x}{(\cos x)\sin x + (\sin x)\cos x} \, dx \] Step 1: Simplify the integrand Notice that the numerator and the denominator have the same terms: \[ (\cos x)\sin x \quad \text{and} \quad (\sin x)\cos x \] Since multiplication is commutative, these terms are identical. Therefore, the integrand simplifies as follows: \[ \frac{(\cos x)\sin x}{(\cos x)\sin x + (\sin x)\cos x} = \frac{(\cos x)\sin x}{2(\cos x)\sin x} = \frac{1}{2} \] Step 2: Set up the integral Now the integral becomes: \[ I = \int_0^{\pi/2} \frac{1}{2} \, dx \] This is a simple integral: \[ I = \frac{1}{2} \int_0^{\pi/2} 1 \, dx \] Step 3: Evaluate the integral The integral of 1 with respect to \(x\) is just \(x\), so: \[ I = \frac{1}{2} \left[ x \right]_0^{\pi/2} \] Evaluating the limits: \[ I = \frac{1}{2} \left( \frac{\pi}{2} - 0 \right) = \frac{\pi}{4} \] Conclusion Thus, the value of the integral is: \[ \boxed{\frac{\pi}{4}} \]