Question

In Exercises \(1-20,\) find \(d y / d x\). $$y=\int_{1}^{x} e^{u} \sec u d u$$

Short answer

\(\frac{d y}{d x} = e^{x} \sec{x}\)

Step-by-step solution

Identify the Integral Function

Notice that the function given is an integral from 1 to \(x\) of the function \(e^{u} \sec u\). The variable u is a dummy variable and x is the upper limit of the integral.

Apply the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus, Part 1, tells us that the derivative of the integral of a function is that function. We can simply plug in the upper limit x directly into the place of u. So, \(\frac{d y}{d x} = e^{x} \sec{x}\) as the solution.