Question

State the Fundamental Thereom of calculus and use it to integrate:
e^(t^2) tan(sqrt(t) dt , from x to x^4.


Please show work
Kind regards...

Answer 1

\[\int\limits_{x}^{x^4} e ^{t ^{2}} \tan \sqrt{t} \]

Answer 2

using fundamental theorem of calculus \[\Large \frac{d}{dx}\int\limits_{v(x))}^{u(x)}f(t)dt=f(u(x))u'(x)-f(v(x))v'(x)\]
here we have
u(x)-x^4
v(x)=x
can you do this ???

Answer 3

@unseenoceans can you do this now?

Answer 4

sorry sami, I have problems with my screen. it goes on and off. I know how to plug in the limits b and I am not sure after that.

Answer 5

okay screen is back on ...lemme see.

Answer 6

\[e ^{x ^{4^{2}}} \tan \sqrt{x ^{4}}.2x - e ^{x ^{2}}-\tan \sqrt{x}.1\]

Answer 7

how does that look sami . does it need more simplification?

Answer 8

erivative of u(x)=x^4 is u'(x)=4x^3 !!!!!!! you wrote 2x !

Answer 9

derivative*

Answer 10

okay...2x should be 4x^3 ...my bad

Answer 11

yeah i just seen that...silly mistake...its 4am in Ireland. ;)

Answer 12

also it is tan and not -tan

Answer 13

hmmm at 4am :P
there is another mistake find it !!

Answer 14

That was a typo. I swear..

Answer 15

ok !
i am too quite famous for typos here ask someone about me :P

Answer 16

lol...so is that the solution then sami?..is it that easy....It looks monstrous but it really is not!

Answer 17

just plug and chug and a bit of chain rule and that's it...Thanks for your help. You've been really great!

Answer 18

you 're welcome:) btw if you want you can take\[\Large e^{x^2}\] common!

Answer 19

yes and (e^x^4^2 whould read e^x^6 ....

Answer 20

after common it should go down to e^x^4

Answer 21

what abou tan(sqrt(x) can I take that too?

Answer 22

it should get to \[\Large e^{x^6}\]
because when you used x^4 at the time of substitution
it should be
\[\Large e^{t^2}=e^{(x^4)^2}\]
\[\Large e^{x^8}\]

Answer 23

yes, you're correct I used the wrong law \[a^{m} \times a^{n} = a ^{m+n} \]

Answer 24

correct one is (a^m)n = a^mn

Answer 25

yes !

Answer 26

Anyway thanks you sami.....time for me to go bed now....I really appreciate your time and effort....good luck and see you around on openstudy my friend!

Answer 27

you are welcome:)