math problem
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how do you integrate (X^2 +1)^0.5 dx
the question is.. prove that the above integration is equal to..
(1/2)X( X^2 + 1)^0.5 + 0.5ln (X + (X^2 +1)^0.5) + C
Do you do integration by parts?
Dammit.. hate university maths. I think it's simple actually... but not for arithmetically challenged individuals like me =( -
edit error!
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1+1=2
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deleted cos careless in first partOriginally posted by sir sickolot II:how do you integrate (X^2 +1)^0.5 dx
the question is.. prove that the above integration is equal to..
(1/2)X( X^2 + 1)^0.5 + 0.5ln (X + (X^2 +1)^0.5) + C
Do you do integration by parts?
Dammit.. hate university maths. I think it's simple actually... but not for arithmetically challenged individuals like me =( -
does your question allow you to differentiate the bottom to get the top instead?
Or die die need to do integration? -
substitution, then simplify and get integrate sec^3x, then by parts and resub back
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differentiate the other side can or not?Originally posted by sir sickolot II:how do you integrate (X^2 +1)^0.5 dx
the question is.. prove that the above integration is equal to..
(1/2)X( X^2 + 1)^0.5 + 0.5ln (X + (X^2 +1)^0.5) + C
Do you do integration by parts?
Dammit.. hate university maths. I think it's simple actually... but not for arithmetically challenged individuals like me =( -
deleted again after discovering another careless mistake.Originally posted by angelfairy:substitution, then simplify and get integrate sec^3x, then by parts and resub back -
Originally posted by tanjun:
secutanu, not sec^3u, and tan^2usecu, not sec^2u(secutanu) if u using U = secu
Thanks for the hint.
Let x = tan u
dx = sec^2 u du
Sub above eqn into original eqn.
Integrate (x^2 + 1)^1/2 = Integrate (tan^2 u + 1)^1/2 (sec^2 u)du
=Integrate (sec^2 u)^1/2 (sec^2 u)du
= Integrate (sec^3 u) du
By parts
Spilt sec^3 u into sec u and sec^2 u. U can choose your u to be either one.
Formula: Integrate u(dv/dx) dx = uv - Integrate v(du/dx) dx
I make U to be sec u.
dU/dx = sec u tan u
= Integrate (sec u)(sec^2 u) du = sec^3 u - integrate sec^2 u(sec u tan u) du
= sec^3 u - 1/3 sec^3 u + c
= 2/3 sec^3 u + c
Drawing a right-angled triangle,
sec u = (x + 1)^2
= 2/3(x + 1)^6 + c
Am i correct? -
Usually proving questions have a pre-question
I wonder if there is any other question parts before this because there's a very much simpler method if there is,
ie before this question, you were asked to prove
integrate (X^2 +1)^-0.5 dx = ln (X + (X^2 +1)^0.5) -
= Integrate (sec u)(sec^2 u) du = sec^3 u - integrate sec^2 u(sec u tan u) du
U = sec u
= sec^3 u - 1/3 sec^3 u + c
= 2/3 sec^3 u + c
dU/dx = sec u tan u
V = sec^2 u
dV/dx = 2sec^2 u tan u
UV - integrate V(dU/dx) = integrate U(dV/dx)
sec^3 u - integrate sec^2 u(sec u tan u)
as stated above.
However, ans is still vastly different from the one given above with a ln function.
Differentiate the above ans.
1/4(1 + x^2)^-1/2 + (2x + 2(x^2 + 1)^1/2 )^-1
Combining these two fractions will make things very complicated. -
Try doing by parts right at the very start without substitution firstOriginally posted by tanjun:U = sec u
dU/dx = sec u tan u
V = sec^2 u
dV/dx = 2sec^2 u tan u
UV - integrate V(dU/dx) = integrate U(dV/dx)
sec^3 u - integrate sec^2 u(sec u tan u)
as stated above.
However, ans is still vastly different from the one given above with a ln function.
Differentiate the above ans.
1/4(1 + x^2)^-1/2 + (2x + 2(x^2 + 1)^1/2 )^-1
Combining these two fractions will make things very complicated.
at least it seems so to me
Calculations are much more simplified
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(1 + x^2)^0.5Originally posted by eagle:Try doing by parts right at the very start without substitution first
at least it seems so to me
Calculations are much more simplified
Very hard to do by parts leh.
Cos it is not a product of two simpler terms. It functions like a prime number with factors 1 and the number itself. -
since the qn gives u the two sides and prove, differentiate the result.
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let dv/dx be 1Originally posted by tanjun:(1 + x^2)^0.5
Very hard to do by parts leh.
Cos it is not a product of two simpler terms. It functions like a prime number with factors 1 and the number itself.
u be (1 + x^2)^0.5
u will be able to reduce the question to integrating only (1 + x^2)^-0.5, which when u do substituition of x=tan u, you only need to integrate sec u. This gives you ln (sec u + tan u), which when converted to x, will give you the answer
edit: careless typo -
actually I tried both sides. It seems to me integration as slightly less tedious in this questionOriginally posted by ^tamago^:since the qn gives u the two sides and prove, differentiate the result.
But I'm still wondering if TS has posted the full question. -
oh God this just reminds me that my math is gg
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Differentiate the above ans.Originally posted by ^tamago^:since the qn gives u the two sides and prove, differentiate the result.
1/4(1 + x^2)^-1/2 + (2x + 2(x^2 + 1)^1/2 )^-1
as stated in above reply.
However, the main problem I think is the different denominator existed in both terms. Even if I combine, it is of no use. One denominator contains 1 + x^2 while the other has a 2x added on to 2(x^2 + 1). Addition of terms is hard to cancel out.
I am doubting the accuracy of the ans. -
the answer is correct. I managed to get itOriginally posted by tanjun:Differentiate the above ans.
1/4(1 + x^2)^-1/2 + (2x + 2(x^2 + 1)^1/2 )^-1
as stated in above reply.
However, the main problem I think is the different denominator existed in both terms. Even if I combine, it is of no use. One denominator contains 1 + x^2 while the other has a 2x added on to 2(x^2 + 1). Addition of terms is hard to cancel out.
I am doubting the accuracy of the ans. -
thanks all and eagle. it's the full question.
the solution i'm very sure does not involve trigo, although it would defintely be easier to solve using trigo.
reason being my maths course does not cover trigo integration
i think integration by parts is my best bet.... or by dy/dx the RHS......
haiz. -
hmmm........ ok I go write out the full solution nicely on paper and take photo with my hp camera now
give me 5 mins -
Yup. I managed to get it too.Originally posted by eagle:the answer is correct. I managed to get it
Oh. I guess qn should give a hint first like use by parts followed by substitution first or sumthing like that.
Cos, as above, I use substitution first, then by parts, ans is 2/3 (1 + x)^6 +c
Ans is vastly different from the other ans (1/2)X( X^2 + 1)^0.5 + 0.5ln (X + (X^2 +1)^0.5) + C which use by parts first then substitution. -
wow thank you thank you!!!! veri touchedOriginally posted by eagle:hmmm........ ok I go write out the full solution nicely on paper and take photo with my hp camera now
give me 5 mins
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scanning now
did both methods -
Thanks for the gd qn. It refreshes my maths concept back in JC days and thanks for the hints and corrections.