Question

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{-\pi}^{x} \frac{2-\sin t}{3+\cos t} d t$$

Short answer

The derivative of the function \(y\) with respect to \(x\) is \(dy/dx = \frac{2-\sin x}{3+\cos x}\).

Step-by-step solution

Identify the Function Inside the Integral

The function inside the integral is \(\frac{2-\sin t}{3+\cos t}\). This is the function we will be using to differentiate \(y\) with respect to \(x\). We also observe that \(x\) is the upper limit of the integral.

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if a function \(y\) is defined as an integral from a constant to \(x\) of a function, then its derivative is simply the function. So we find the derivative \(dy/dx\) by replacing \(t\) with \(x\) in the function inside the integral. In this case, we obtain \(dy/dx = \frac{2-\sin x}{3+\cos x}\).

Simplify the Expression

In this case, there isn't anything that needs to be simplified. Hence, \(dy/dx = \frac{2-\sin x}{3+\cos x}\) is the final answer.