integral from 0 to pi/4 of (sec(x))^4 dx

[cheffy]

integral from 0 to pi/4 of (sec(x))^4 dx

I got it down to the integral of 1/(cos(x)^2)(1-(sin(x)^2) but I don't know what to do next. Should I replace the other cos(x)^2 too?

Thanks!

 

[galactus]

Use the reduction formula:

\(\displaystyle \L\\\int{sec^{n}(x)}dx=\frac{sec^{n-2}(x)tan(x)}{n-1}+\frac{n-2}{n-1}\int{sec^{n-2}(x)}dx\)

 

[cheffy]

Is that an identity that I'm just supposed to know or does it derive from somewhere?

 

[galactus]

It is a reduction formula and should be in your calc book. It can be derived.

Here we go without the formula.

\(\displaystyle \L\\\int{sec^{4}(x)}dx\\=\int(1+tan^{2}(x))sec^{2}(x)dx\\=\int(sec^{2}(x)+tan^{2}(x)sec^{2}(x))dx\\=tan(x)+\frac{1}{3}tan^{3}(x)+C\)