Question

Evaluate: \[ \int_0^{\frac{\pi}{2}} \frac{5 \sin x + 3 \cos x}{\sin x + \cos x} \, dx \]

Hint:

When faced with sums of sine and cosine functions in the denominator, look for substitution that can simplify the expression, potentially reducing the complexity of the integral.

Answer 1

First, break the integrand into two parts: \[ \int_0^{\frac{\pi}{2}} \frac{5 \sin x + 3 \cos x}{\sin x + \cos x} \, dx = \int_0^{\frac{\pi}{2}} \left( 5 \frac{\sin x}{\sin x + \cos x} + 3 \frac{\cos x}{\sin x + \cos x} \right) dx \] Now, observe that both integrals have the form \( \frac{\sin x}{\sin x + \cos x} \) and \( \frac{\cos x}{\sin x + \cos x} \), which can be simplified by substitution. Let: \[ u = \sin x + \cos x \quad \Rightarrow \quad du = (\cos x - \sin x) \, dx \] After performing the necessary simplifications and substitutions, the final solution is: \[ \int_0^{\frac{\pi}{2}} \frac{5 \sin x + 3 \cos x}{\sin x + \cos x} \, dx = 5 \ln(\sin x + \cos x) \Big|_0^{\frac{\pi}{2}} = 5 \ln(1) - 5 \ln(1) = 0 \]