Problem 39 - Essential Calculus

Question

Use the tabular method to find the integral. $$ \int x \sec ^{2} x d x $$

Step-by-Step Solution

Answer

The integral of \(x \sec^2 x dx\) is \(x \tan x + \ln |\cos x| + C\).

Step 1: Identify u and dv

Firstly, identify parts of the integrand to match the integration by parts formula. Choose \(u = x\) and \(dv = \sec^2 x dx\). This choice is made because the integral of \(\sec^2 x\) (which is \(tan x\)) is simpler than the original integrand.

Step 2: Compute du and v

Calculate \(du\) and \(v\) based on our choices of \(u\) and \(dv\). The derivative of \(u = x\) is \(du = dx\), and the integral of \(dv = \sec^2 x dx\) is \(v = \tan x\).

Step 3: Apply integration by parts formula

Use the integration by parts formula \(\int u dv = uv - \int v du\). Substituting \(u\), \(v\), \(dv\), and \(du\) from the previous steps generates \(\int x \sec^2 x dx = x \tan x - \int \tan x dx\).

Step 4: Integrate \(v du\)

Compute the outstanding integral \(\int \tan x dx\). The antiderivative of \(\tan x\) is \(-\ln |\cos x|\). Thus, the original integral becomes \(x \tan x + \ln |\cos x|\). Add the constant of integration \(C\) to make sure all possible antiderivatives are included, so the result is \(x \tan x + \ln |\cos x| + C\).