Problem 41 - Essential Calculus

Question

Find the indefinite integral. $$ \int \tan ^{4} x \sec ^{2} x d x $$

Step-by-Step Solution

Answer

The integral of \(\tan ^{4} x \sec ^{2} x \, d x\) is \(\frac{\tan^{5} x}{5} + C\).

Step 1: Identifying the Substitution

You can see the integrand includes \(\tan ^{4} x\) and \(\sec ^{2} x\). Since \(\sec ^{2} x\) is the derivative of \(\tan x\), it is a good idea to substitute \(u = \tan x\). As a result, \(du = \sec ^{2} x \, dx\).

Step 2: Substituting in the Integral

Substitute \(u = \tan x\) and \(du = \sec ^{2} x \, dx\) into the integral, we get \(\int u^{4} \, du\).

Step 3: Integrating

The integral \(\int u^{4} \, du\) is a simple power rule integral. The rule states that \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\). After applying the power rule, we get \(\frac{u^{5}}{5} + C\).

Step 4: Replacing u with original function

Lastly, substitute back \(u = \tan x\), so the solution is \(\frac{\tan^{5} x}{5} + C\).