Question

$\int {\sec }^{3}xdx$

• A. $1/2[\sec x\tan x-\log |\sec x+\tan x|]+C$
• B. $1/2[\sec x\tan x+\log |\sec x-\tan x|]+C$
• C. $1/2[\sec x\tan x-\log |\sec x-\tan x|]+C$
• D. $1/2[\sec x\tan x+\log |\sec x+\tan x|]+C$

Answer 1

Answer: (D)

Let $I=\int {\sec }^{3}xdx$
$=\int \underset{I}{\sec }x{\underset{II}{\sec }}^{2}xdx$ $\sec x\tan x-\int \sec xta{n}^{2}xdx$ (using integration by parts)
$=\sec x\tan x-\int \sec x({\sec }^{2}x-1)dx$
$\Rightarrow$ $I=\sec x\tan x-\int {\sec }^{3}xdx+\int secxdx$
$\Rightarrow$ $I=\sec x\tan x-I+\log |secx+tanx|+2C$
$\Rightarrow$ $2I=\sec x\tan x+\log |secx+\tan x|+2C$
$\Rightarrow$ $I=\frac{1}{2}[\sec x\tan x+\log |\sec x+\tan x|]+C$