48. Integral of sec³x Use integration by parts to show that:
∫ sec³x dx = (1/2) secx tanx + (1/2) ∫ secx dx

Verified step by step guidance

1. Start with the integral \( \int \sec^3 x \, dx \). Rewrite \( \sec^3 x \) as \( \sec x \cdot \sec^2 x \) to prepare for integration by parts.
2. Choose \( u = \sec x \) and \( dv = \sec^2 x \, dx \). Then compute \( du = \sec x \tan x \, dx \) and \( v = \tan x \) since the derivative of \( \tan x \) is \( \sec^2 x \).
3. Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the chosen \( u \), \( v \), \( du \), and \( dv \) to get \( \int \sec^3 x \, dx = \sec x \tan x - \int \tan x \cdot \sec x \tan x \, dx \).
4. Simplify the integral \( \int \tan x \cdot \sec x \tan x \, dx = \int \sec x \tan^2 x \, dx \). Use the identity \( \tan^2 x = \sec^2 x - 1 \) to rewrite the integral as \( \int \sec x (\sec^2 x - 1) \, dx = \int \sec^3 x \, dx - \int \sec x \, dx \).
5. Substitute back into the equation and solve for \( \int \sec^3 x \, dx \) to isolate it on one side. This will lead to the expression \( \int \sec^3 x \, dx = \frac{1}{2} \sec x \tan x + \frac{1}{2} \int \sec x \, dx \), as required.

Key Concepts

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing appropriate u and dv is crucial to simplify the integral effectively.

Trigonometric Identities

Trigonometric identities relate different trigonometric functions and are essential for simplifying integrals involving trig functions. For example, knowing that d/dx (sec x) = sec x tan x helps in recognizing parts of the integral and applying integration by parts correctly.

Integral of sec x

The integral of sec x, ∫ sec x dx, is a standard integral that equals ln |sec x + tan x| + C. Recognizing this integral within the problem allows for expressing the solution in terms of known functions, completing the integration process.