Question

Which strategy shall I use to evaluate this integral?

Answer 1

\[\large \int\limits_{}^{}\sec^3x~dx\]

Answer 2

Hint:Write sec^3x like this-> (1+tan^2x)secx

Answer 3

I don't see anything I can substitute there after writing it like that.

Answer 4

or you could use a reduction formula

Answer 5

Hmm, what is the reduction formula??

Answer 6

Try integrating by parts with
\[\begin{matrix}
u=\sec x&&&\mathrm dv=\sec^2x\,\mathrm dx\\[1ex]
\mathrm du=\sec x\tan x\,\mathrm dx&&&v=\tan x
\end{matrix}\]Then
\[\begin{align*}
\int\sec^3x\,\mathrm dx&=\sec x\tan x-\int\sec x\tan^2x\,\mathrm dx\\[1ex]
&=\sec x\tan x-\int\sec x(\sec^2x-1)\,\mathrm dx\\[1ex]
2\int\sec^3x\,\mathrm dx&=\sec x\tan x+\int\sec x\,\mathrm dx
\end{align*}\](This is the result of the reduction formula which is derived from integrating by parts.)

Answer 7

\[\sec(x)=\sec(x)\frac{\sec(x)+\tan(x)}{\sec(x)+\tan(x)}=\frac{\sec^2(x)+\sec(x)\tan(x)}{\sec(x)+\tan(x)}\]

Answer 8

Wow, I would never have guessed that this was a integration by parts problem. But that makes it so much easier.

Answer 9

Thank you guys for the help.