Need some help

[HappyCalculusStudent]

I am having a lot of trouble calculating the definite integral

?(sec(t/2))^4 dt (from 0 to ?/2)

Anyway, I get different answers:
8/3 when I use substitution,
and
2 using a table of integrals - reduction formula for ?(sec(u))^n du
and I even get
4/3 ...

Could someone please show me step by step how to do this to get the correct answer?
Thank you!

(NOTE: If the symbols don't display properly, that's supposed to be an integral sign. Also, the final fraction has "pi" as the numerator.)

 

[galactus]

Try breaking your sec up:

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

 

[Deleted member 4993]

I am having a lot of trouble calculating the definite integral

?(sec(t/2))^4 dt (from 0 to ?/2)

Anyway, I get different answers:
8/3 when I use substitution,
and
2 using a table of integrals - reduction formula for ?(sec(u))^n du
and I even get
4/3 ...

Could someone please show me step by step how to do this to get the correct answer?
Thank you!

(NOTE: If the symbols don't display properly, that's supposed to be an integral sign. Also, the final fraction has "pi" as the numerator.)

\(\displaystyle \int_0^{\frac{\pi}{2}}[\sec(\frac{t}{2})]^4dt\)

first substitute:

\(\displaystyle u \, = \frac{t}{2}\)

\(\displaystyle dt \, = \, 2\cdot u\)

\(\displaystyle \int_0^{\frac{\pi}{2}}[\sec(\frac{t}{2})]^4dt\)

\(\displaystyle =2\cdot \int_0^{\frac{\pi}{4}}[\sec(u)]^4du\) .... notice the change in limits and a factor of 2 in front.

\(\displaystyle =2\cdot \int_0^{\frac{\pi}{4}}\sec^2(u)\cdot [1 \, + \, \tan^2(u)]du\)

Now continue... the answer should be 8/3

 

[HappyCalculusStudent]

Re: Need some help - using tables, too

Thank you very much for your help!
The only thing I still don't understand is why I got the other answers. I'm not comfortable applying the formulas from tables.

Given the reduction formula
?(sec(u))^n du = 1/(n-1) tan(u)(sec(u))^(n-2) + (n-2)/(n-1)?(sec(u))^(n-2) du

I let n=4 and u=(t/2)
1/3 tan(u)(sec(u))^2 + 2/3?(sec(u))^2 du
changing the limits of integration??? 0 to ?/4
and substituting, I got
1/3 tan(?/4)(sec(?/4))^2 + 2/3 tan (?/4)
=1/3 (1)(2) + 2/3 = 2/3 + 2/3 = 4/3

The other time I did the first part the same.
I let n=4 and u=(t/2)
1/3 tan(u)(sec(u))^2 + 2/3?(sec(u))^2 du
but I calculated the latter integral, substituting u=t/2.
Then 2du=dt and
1/3 tan(t/2)(sec(t/2))^2 + 2/3*2(tan(t/2)
evaluating from 0 to?/2
=1/3 (1)(2) + 4/3 = 2/3 + 4/3 = 2

So I'd like to learn to use the tables properly when u is a fraction or multiple of the variable. I'd appreciate your advice.
Thank you again!