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.Sam.
 one year ago
Best ResponseYou've already chosen the best response.1Hmm, try substitute \[u=\cos^2(x) \\ \\du=2\cos(x)\sin(x)\] \[\int\limits \frac{1}{\sqrt{u+1}} \, du\]\ Substitute one more time, \[t=u+1 \\ \\dt=du\] \[\int\limits \frac{1}{\sqrt{t}} \, dt\] Can you do it now?

.Sam.
 one year ago
Best ResponseYou've already chosen the best response.1Note: from starting, I used sin(2x)=2sin(x)cos(x)

shaqadry
 one year ago
Best ResponseYou've already chosen the best response.0i dont really get it :(

.Sam.
 one year ago
Best ResponseYou've already chosen the best response.1Thought so, :D I'll rearrange it

shaqadry
 one year ago
Best ResponseYou've already chosen the best response.0in my book it says that the answer is 2 √(1 + cos² x) + c

.Sam.
 one year ago
Best ResponseYou've already chosen the best response.1You have \[\int\limits \frac{\sin 2x }{\sqrt{ \cos ^2x+1}} \, dx\] Change the sin(2x) to 2sin(x)cos(x), That's an identity. \[\int\limits \frac{2sin(x)cos(x) }{\sqrt{ \cos ^2x+1}} \, dx\] Then, let \[u=\cos^2(x) \\ \\du=2\cos(x)\sin(x) dx \\ \\du=2\sin(x)\cos(x)dx\] \[\int\limits \frac{du }{\sqrt{ u+1}} \, \] Substitute one more time, Let \[t=u+1 \\ \\dt=du\] \[\int\limits \frac{dt }{\sqrt{ t}} \, \] \[\int\limits \frac{1 }{\sqrt{ t}} dt\, \] \[\int\limits t^{\frac{1}{2}}dt\, \] Integrate it, \[2 \sqrt{t}+c\] Substitute back, \[2 \sqrt{u+1}+c\] \[2 \sqrt{cos^2x+1}+c\]

shaqadry
 one year ago
Best ResponseYou've already chosen the best response.0i see... but in my level, we haven't learn how to differentiate cos² x directly, we only learn how to differentiate cos x so i dont really get it is there any other way?

.Sam.
 one year ago
Best ResponseYou've already chosen the best response.1I don't think so, but differentiating cos² x is using chain rule, you should know that before doing integration. Hmm, that's weird.

shaqadry
 one year ago
Best ResponseYou've already chosen the best response.0okay i'll try. thank you!

shaqadry
 one year ago
Best ResponseYou've already chosen the best response.0i only didnt undertsand that part but the rest i do so thanks a lot !
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