U-substitution for trigonometric functions

Derek was told to evaluate the following problem: $\displaystyle \int \sec^2 x \tan x \, dx.$

Which, if any, of Derek's three steps was incorrect?

Step 1:

• $u = \sec x$
• $du = \sec x \tan x\, dx$

Step 2:

• $dx = \dfrac{du}{\sec x \tan x}$
• $\displaystyle \int \frac{\sec^2 x \tan x}{\sec x \tan x} du$
• $\displaystyle \int u\, du$

Step 3:

• $\displaystyle \int u\, du = \frac{u^2}{2} + C$
• $\displaystyle \int \big( \sec^2 x \tan x\big)\, dx = \frac{\sec^2 x}{2} + C$