Question

Evaluate the Indefinite Integral.

Answer 1

\[\int\limits_{}^{}\sec^4(x/2)\]

Answer 2

Looks like you're going to need a couple of trig identities here: one for the HALF angle (x/2) and at least one for the 4th power of the secant function. If it were I, I'd focus on breaking the 4th power of the sec. up so that it would be more easily integrated. Then worry about the half angle.

Answer 3

What half angle formula would I use?

Answer 4

i would start with \[2\int \sec^4(u)du\] a mental u-sub
then look in the back of the book for the reduction formula
it is there

Answer 5

\[I=\int\limits \sec ^4(\frac{ x }{ 2 })dx\]
put \[\frac{ x }{ 2 }=u,x=2u,dx=2du\]
\[I=2 \int\limits \sec ^4 du=2 \int\limits( \sec ^2u)(\sec ^2u)du=2 \int\limits \left( 1+\tan ^2u \right)\sec ^2u~du\]
\[=2\left[ \int\limits \sec ^2u~du+\int\limits \tan ^2u~\sec ^2u~du \right]+c\]
......

Answer 6

I believe there is a formula for sec (x/2), that is, for the secant of HALF the angle x. Look for it. Half angle formulas for the sine, cosine and tangent functions are more commonly seen.

Answer 7

\[=2\left[ \tan u+\frac{ \tan ^3u }{ 3 } \right]+c\]
replace the value of u by x/2