Question

$\int {\sec }^{3}xdx$ is equal to

• A. $\sec x\tan x+\log |\sec x+\tan x|+c$
• B. $\frac{1}{2}\sec x\tan x+\frac{1}{2}\log |\sec x+\tan x|+c$
• C. ${\sec }^{3}x\tan x+\log |{\sec }^{3}x+\tan x|+c$
• D. None of the above

Answer 1

Answer: (B)

Let $I=\int {\sec }^{2}xdx$
$\Rightarrow$ $I=\int \sec x\cdot {\sec }^{2}xdx$
$\Rightarrow$ $I=\sec x\tan x-\int \sec x\tan x\cdot \tan xdx$ (integrating by parts) $I=\sec x\tan x-\int \sec x{\tan }^{2}xdx$
$=\sec x\tan x-\int \sec x({\sec }^{2}x-1)dx$
$\Rightarrow$ $I=\sec x\tan x-\int ({\sec }^{2}x-\sec x)dx$
$\Rightarrow$ $I=\sec x\tan x-\int {\sec }^{2}xdx+\int \sec xdx$
$\Rightarrow$ $2I=\sec x\tan x+\int \sec xdx$
$\Rightarrow$ $2I=\sec x\tan x+\log |\sec x+\tan x|+c$
$\Rightarrow$ $I=\frac{1}{2}\sec x\tan x+\frac{1}{2}\log |\sec x+\tan x|+c$