Question

integral sqrt(4t^2 +16t^6) dt from 0 to 2

Answer 1

i got upto integral sec^3 x

Answer 2

rewrite it in simpler form is my first thought

Answer 3

\[\sqrt{4t^2(1+4t^4)}=2t\sqrt{1+(2t)^2}\]
maybe?

Answer 4

this can be integrated by parts if i see it right

Answer 5

careful A, your equation above isn't quite right.

Answer 6

\begin{array}c
&&\sqrt{1+(2t^2)^2)}\\
+&2t&\int^1 v\\
-&2&\int^2v\\
+&0&---
\end{array}

yep, maybe a better setup, but im not sure if this is the right track to go down yet

Answer 7

\[ 2t \sqrt{1+4t^4} \]
Now substitute \( u = t^2 \) or \( u = 2t^2 \) if you prefer.

Answer 8

I would use a substitution 2t^2=tan(u).

Answer 9

most integrals are unsolvable, so I wouldnt worry to much about it ;)

Answer 10

it is just very lengthy

Answer 11

We rewrite the expression as
\[I=\int\limits_{}^{}2t \sqrt{1+(2t^2)^2}dt\]
By substituting \(2t^2=\tan{u} \implies 4tdt=\sec^2(u)du \implies dt=\frac{\sec^2u}{4t}du\), we get:
\[I={1 \over 2}\int\limits \sqrt{1+\tan^2u}\sec^2{u}du={1 \over 2}\int\limits \sec^3{u}du.\]

Answer 12

i did integration by parts and got 1/2(secxtanx +ln(sec+tanx) from 0 to 8

Answer 13

@ Mr. Math, yes i did that and got integral 1/2 sec^3 x

Answer 14

i was looking for a way to do it in less steps.

Answer 15

Hmm, I don't have such a way right now, sorry :D

Answer 16

less steps? ... cant see it in lesser

Answer 17

By the way, the integration of [sec(u)]^3 is not an easy one. You will need to use integration by parts or some other technique.

Answer 18

hm. i hope i dont get this on exam

Answer 19

thank you everyone for helping though:)

Answer 20

You can do it if you get it on your exam!

Answer 21

Couldn't agree more with Mr M. Work this problem through. It's a great exercise. And just because it requires more steps than usual should be thought of as a challenge to master rather than an impossibility.

I remember when I was at your stage I'd take harder integrals like this and redo them once or twice a day for several days until I had it down cold. Sometimes I wish I still was that kid. ;-)

Answer 22

Lol, you sounded like an old man :P

Answer 23

Either that or I'm still a bit hung over from last night and remembering when that didn't happen.