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

Guida verificata passo dopo passo

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.

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 \( \sec^2 x \).

Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the chosen \( u \), \( v \), \( du \), and \( dv \) to get \( \int \sec^3 x \, dx = \sec x \tan x - \int \tan x \cdot \sec x \tan x \, dx \).

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 \( \int \sec x (\sec^2 x - 1) \, dx = \int \sec^3 x \, dx - \int \sec x \, dx \).

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.

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