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Chapter 17: Problem 11

Use integration by parts to find \(\int \sec ^{3} x \mathrm{~d} x\).

Short Answer

Expert verified
Answer: The integral of \(\sec^3 x\) with respect to x is \(\int \sec^3 x dx = (\sec^2 x) \ln|\sec x + \tan x| - 2\int \sec x \tan x \ln|\sec x + \tan x| dx + C\), where C is the constant of integration.

Step by step solution

01

Choose u and v'

Select u and v' for integration by parts, using the LIATE rule. In this problem, u will be \(\sec^{2}x\) and v' will be \(\sec x\).
02

Find the derivatives and integrals

Calculate the derivative of u, and integral of v': - Calculate u': Differentiate u with respect to x: \(u' = \frac{d}{dx} (\sec^{2}x) = 2\sec x\tan x \) - Calculate v: Integrate v' with respect to x: \(v = \int \sec x dx = \ln|\sec x + \tan x|\)
03

Apply the integration by parts formula

Using the integration by parts formula, we have: \(\int \sec^{3} x dx = \int u v' dx = u v - \int v u' dx \) We now substitute u, u', v, and v' from Steps 1 and 2: \(\int \sec^{3} x dx = (\sec^{2}x)\left(\ln|\sec x + \tan x|\right) - \int \left(\ln|\sec x+\tan x|\right)(2\sec x\tan x)dx\)
04

Simplify the expression

We can simplify expression: \(\int \sec^{3} x dx = (\sec^{2}x) \ln|\sec x+\tan x| - 2\int \sec x\tan x \ln|\sec x+\tan x| dx \) Now the integral is in a simpler form with a reduced power of \(\sec x\).
05

Result

The integral of \(\sec^{3}x\) with respect to x is: \(\int \sec^{3} x dx = (\sec^{2}x) \ln|\sec x+\tan x| - 2\int \sec x\tan x \ln|\sec x+\tan x| dx + C\) Where C is the constant of integration.

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