Integrate x^5secxdx to Solving the Tricky Integration Problem

[islubio]

Hi I m kinda stuck at this question.

integral of x^5secxdx. I tried IBP n i couldn't carry on.

 

[Dickfore]

Unfortunately, this integral is not expressible in terms of elementary functions.

 

[islubio]

Ok my frene gave me the wrong info. int that interms of -1 to 1.
I know the answer is 0 but why?

 

[moxy]

Is the function $f(x) = x^5 \sec{x}$ even, odd or neither? When you have decided that, it should become clear as to why

$\int^{1}_{-1}{x^5 \sec{x}} dx = 0$

 

[islubio]

I feel lost lol. How do i tell if it's even odd or neither?

 

[moxy]

A function $f$ is even if $f(x) = f(-x) \forall x \in domain$

A function $f$ is odd if $-f(x) = f(-x) \forall x \in domain$

For your function, check $f(x)$ vs. $f(-x)$ vs. $-f(x)$.

 

[islubio]

f(-x) = -x^5sec-x

-f(x) = -(x^5secx)

Is this the way?

 

[moxy]

Sorry, I initially missed a negative sign in my last post.

Yes, that's a way to check. And you know that sec(x) = 1/cos(x) and that cos(-x) = cos(x). Therefore, sec(-x) = sec(x). That implies that sec is an even function. However, (-x)^5 = -(x^5) implies that x^5 is an odd function. An even function times an odd function is itself an odd function.

One property of an odd function, g(x), is that

$\int^{a}_{-a} g(x)dx = 0$

 

[moxy]

On a related note, for an even function h(x),

$\int^{a}_{-a} h(x)dx = 2\int^{a}_{0} h(x)dx$

If you graph some simple even and odd functions, you will see why this is the case.