$$ \int_0^{\frac{\pi}{2}} \frac{3 \sin x+4 \cos x}{\sin x+\cos x} d x= $$
(A) $$ \frac{\pi}{4} $$
(B) $$ \frac{7\pi}{4} $$
(C) $\pi$
(D) $$ \frac{7\pi}{2} $$

Question

Accepted Answer

> [!TLDR]
> The integral simplifies by splitting the numerator into a linear combination of the denominator and its derivative, leading to a sum of a constant and a logarithmic term; the final value is $\frac{7\pi}{4}$, so the correct option is **(B)**.

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The key idea is that when you have an integrand of the form $\frac{a\sin x + b\cos x}{c\sin x + d\cos x}$, you can often rewrite the numerator as $A(\text{denominator}) + B(\text{derivative of denominator})$. This works because the derivative of $\sin x + \cos x$ is $\cos x - \sin x$, and any linear combination of $\sin x$ and $\cos x$ can be expressed as a combination of these two. Once you do that, the integral splits into a simple constant term and a logarithmic term that often vanishes over symmetric limits.

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1. **Set up the decomposition**  
   We want constants $A$ and $B$ such that  
   $$
   3\sin x + 4\cos x = A(\sin x + \cos x) + B(\cos x - \sin x).
   $$  
   Expanding the right side:  
   $$
   A\sin x + A\cos x + B\cos x - B\sin x = (A - B)\sin x + (A + B)\cos x.
   $$  
   Matching coefficients with $3\sin x + 4\cos x$ gives the system:  
   $$
   \begin{cases}
   A - B = 3 \
   A + B = 4
   \end{cases}
   $$  
   Adding the equations: $2A = 7 \Rightarrow A = \frac{7}{2}$.  
   Subtracting: $2B = 1 \Rightarrow B = \frac{1}{2}$.

2. **Rewrite the integral**  
   The integrand becomes:  
   $$
   \frac{3\sin x + 4\cos x}{\sin x + \cos x} = \frac{\frac{7}{2}(\sin x + \cos x) + \frac{1}{2}(\cos x - \sin x)}{\sin x + \cos x}
   = \frac{7}{2} + \frac{1}{2} \cdot \frac{\cos x - \sin x}{\sin x + \cos x}.
   $$  
   So the integral is:  
   $$
   I = \int_0^{\pi/2} \frac{7}{2} \, dx + \frac{1}{2} \int_0^{\pi/2} \frac{\cos x - \sin x}{\sin x + \cos x} \, dx.
   $$

3. **Evaluate the first part**  
   $$
   \int_0^{\pi/2} \frac{7}{2} \, dx = \frac{7}{2} \cdot \frac{\pi}{2} = \frac{7\pi}{4}.
   $$

4. **Evaluate the second part**  
   Notice that the numerator $\cos x - \sin x$ is exactly the derivative

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