Question

Complete the table by identifying \(u\) and \(d u\) for the integral. $$ \int f(g(x)) g^{\prime}(x) d x \quad \underline{u=g(x)} \quad \underline{d u=g^{\prime}(x) d x} $$ $$ \int \sec 2 x \tan 2 x d x $$

Short answer

The variable \(u\) in this integral is \(u = 2x\) and the differential \(du\) is \(du = \tan(2x)dx\). The integral in terms of \(u\) is \(\int \sec u du\).

Step-by-step solution

Identify the function inside the integral

First, observe the integral and notice that the secant and tangent are both functions of the variable \(2x\). This indicates that \(2x\) is the function being operated on. Thus, we take \(u = 2x\).

Identify the differential \(du\)

The differential \(du\) is found by taking the derivative of \(u\) with respect to \(x\). In this case, \(u = 2x\), so the derivative of \(2x\) with respect to \(x\) is \(2\). Therefore, \(du = 2dx\). However, in our integral, the \(dx\) is being multiplied by \(\tan(2x)\), which is also \(du\). So, we take \(du = \tan(2x)dx\).

Rewrite the integral in terms of \(u\)

Substitute \(u\) and \(du\) into the integral, and rewrite the integral in terms of \(u\) instead of \(x\). This gives the integral \(\int \sec u du\).