what is the antiderivative of (x+x^2+2)^.5?

Question

I need some help figuring this out...please include the steps to get to your final answer. Thanks!

by the way, i have the answer and it's an exact value (not one rounded off with a calculator number)...i just can't get it without using my calculator...

.5

as in POINT 5

or ONE HALF

(square root)

Answer 1

f = (x+x^2+2)^.5
= [x^2 + 2 . x . 1 + 1^2 + 1]^.5
= [(x + 1)^2 + 1]^.5

Now integration
Let x + 1 = tan(theta)
Note: sec(theta) = [x^2 + 2x + 2]^.5
differentiating
dx + 0 = sec^2(theta) d(theta)

I = Integration of f = Integration{fdx}
= Integration{[tan^2(theta) + 1]sec^2(theta) d(theta)}
= Integration{sec^2(theta) . sec^2(theta) d(theta)}
= Integration{sec^3(theta) d(theta)}

[Applying rule of multiple >> integration (u . v)>> u = sec(theta) and v = sec^2(theta) and please note that Integration sec^2 = tan and integration sec = ln(sec + tan)]

= sec(theta) tan(theta) + ln[sec(theta) tan(theta)] - Integration{sec^3(theta) d(theta)}
= sec(theta) tan(theta) + ln[sec(theta) tan(theta)] - I
So
I = sec(theta) tan(theta) + ln[sec(theta) tan(theta)] - I

Transposing and solving for I,
I = 1/2{sec(theta) tan(theta) + ln[sec(theta) + tan(theta)]}
Substituting the values of sec(theta) and tan(theta)
I = 1/2{(x+1)sqrt(x^2 + 2x + 2) + ln[sqrt(x^2 + 2x + 2) + (x+1)] } + c
where c is constant of integration

Answer 2

Is that exponent 5 or ,5?
If it's ,5, then you want integral (sqrt(x^2 + x + 2))
Complete the square on the (x^2 + x + 2) and write it as
sqrt ((x+ (1/2)^2 + (7/4))
Then it would be a trig substitution. Sinh I think.
Or do you mean the exponent is 5?
I would still complete the square on it to write it in the form
((x+(1/2))^2 + (7/4))^5
expand it with the binomaial theorem, and integrate term by term.

Answer 3

fist we have this (x+x^2+2)^1/2 and then (x+x^2+2)^1/2+1 so we get (x+x^2+2)^3/2 and then (1/(3/2))(x+x^2+2)^3/2 and we have (2/3)(x+x^2+2)^3/2 but my worry is the part inside (x+x^2+2) don't we need to do its antiderivative ??? so find the answer and tell me too

Answer 4

you may expand it and integrate it term by term

Answer 5

(x^2 + x + 2)^(1/2)
((x + 1)^2)^(1/2)
x + 1

ANS : 1