Question

$\int_{0}^{\frac{\pi}{2}}\frac{300 \sin x+100 \cos x}{\sin x+\cos x}dx=...$

• A. $100\pi$
• B. $300\pi$
• C. $200\pi$
• D. $150\pi$

Hint:

For $\int_0^{\pi/2} \frac{a \sin x + b \cos x}{\sin x + \cos x} dx$, the result is $\frac{(a+b)}{2} \cdot \frac{\pi}{2}$.

Answer 1

The Correct Option is A

Step 1: Concept
Use the property $\int_0^a f(x)dx = \int_0^a f(a-x)dx$.

Step 2: Analysis
- $I = \int_{0}^{\pi/2} \frac{300 \sin x + 100 \cos x}{\sin x + \cos x} dx$ - Applying property: $I = \int_{0}^{\pi/2} \frac{300 \cos x + 100 \sin x}{\cos x + \sin x} dx$

Step 3: Calculation
- Adding the two equations: $2I = \int_{0}^{\pi/2} \frac{400(\sin x + \cos x)}{\sin x + \cos x} dx$ - $2I = \int_{0}^{\pi/2} 400 dx = 400 [x]_0^{\pi/2} = 400(\pi/2) = 200\pi$. - $I = 100\pi$.

Step 4: Conclusion
Hence, the value is $100\pi$. Final Answer: (A)