Final answer:
The student's question involves integrating a combination of cosine and sine functions. Each function is integrated separately, resulting in 3sin(x/3) + (2/5)cos(5x/6) + C, where C is the constant of integration.
Step-by-step explanation:
The question asked requires evaluating the indefinite integral of a combination of cosine and sine functions: ∫cos(x/3) - sin(5x/6)dx. To tackle this, we integrate each term separately using basic integration rules. The integration of cosine results in a sine function, whereas the integration of sine yields a negative cosine function.
The integral of cos(x/3) with respect to x is 3sin(x/3) since the chain rule of differentiation tells us that we would need to multiply by 1/3 when differentiating sin(x/3) to get back to cos(x/3). Similarly, the integral of -sin(5x/6) with respect to x is (2/5)cos(5x/6) because the chain rule will require us to multiply by 5/6 when differentiating -cos(5x/6) to get back to -sin(5x/6).
Hence, integrating the given expression term-by-term, we get 3sin(x/3) + (2/5)cos(5x/6) + C, where C represents the constant of integration.