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Integral of x*sqrt(x+1)

ss7

The book says it HAS to be done by parts, but i just can't seem to do it...

Can someone enlighten me?

d/dx (x) = 1.

OP

could you show the whole problem? i keep arriving at odd answers...

Reply 3

marcusmerehay

17

(i'm no good with LaTeX)
[Int(f(x))]

Int(v.du) = uv - Int(u.dv) + constant

v = x du = (x+1)^(1/2)
dv = 1 u= 2/3(x+1)^(3/2)

hence the integral is

2/3x(x+1)^(3/2) - Int(2/3(x+1)^3/2) + k

2/3x(x+1)^(3/2) - 4/15(x+1)^5/2) + k

if i remember correctly.

Hope that helps.

Reply 4

ss7

OP

thats the answer i keep getting but it is apparently:

2/15(3x-2) (x+1)^3/2

Reply 5

marcusmerehay

17

That's strange. Sorry i can't help more :s-smilie:

(this is from an A-grade Maths and FMaths student too :s-smilie:

)

Reply 6

ss7

OP

marcusmerehay

That's strange. Sorry i can't help more :s-smilie:

(this is from an A-grade Maths and FMaths student too :s-smilie:

)

ah =/ Dang.

Reply 7

marcusmerehay

17

i might have a try when i come home from partying, see if i notice it then. if i work it out then i'll post it on here for you :p:

Reply 8

Dogkicker91

1

textbooks wrong, what your doing is right

Reply 9

SimonM

18

Dogkicker91

textbooks wrong, what your doing is right

The textbook isn't wrong, perhaps the answers are the same up to a constant (I can't be bothered to check)

Reply 10

Dadeyemi

13

ss7

thats the answer i keep getting but it is apparently:

2/15(3x-2) (x+1)^3/2

your both right

... factorize...

2/3x(x+1)^(3/2) - 4/15(x+1)^5/2) + k
=(x+1)^(3/2)(2x/3- 4(x+1)/15) + k
=(x+1)^(3/2)((10x- 4(x+1))/15)) + k
=(x+1)^(3/2)((10x- 4x-4))/15)) + k
=(x+1)^(3/2)((6x-4))/15)) + k
=(2/15)(3x-2)(x+1)^(3/2) + k

:wink:

Reply 11

ss7

OP

Dadeyemi

your both right

... factorize...

2/3x(x+1)^(3/2) - 4/15(x+1)^5/2) + k
=(x+1)^(3/2)(2x/3- 4(x+1)/15) + k
=(x+1)^(3/2)((10x- 4(x+1))/15)) + k
=(x+1)^(3/2)((10x- 4x-4))/15)) + k
=(x+1)^(3/2)((6x-4))/15)) + k
=(2/15)(3x-2)(x+1)^(3/2) + k

:wink:

Reply 12

marcusmerehay

17

i had a feeling it was to do with factorising :p:

i just couldn't be bothered to do it :p:

Reply 13

nuodai

17

Let

u = x, \frac{dv}{dx} = \sqrt{x + 1}

so that

\frac{du}{dx} = 1, v = \frac{2}{3}(x + 1)^{\frac{3}{2}}

, then:

\displaystyle\newline[br]\int x\sqrt{x + 1} dx = \frac{2}{3}x(x+1)^{\frac{3}{2}} - \int \frac{2}{3}(x + 1)^{\frac{3}{2}} dx \newline[br]= \frac{2}{3}x(x+1)^{\frac{3}{2}} - \frac{4}{15}(x + 1)^{\frac{5}{2}} + k\newline[br]= \frac{2}{3}(x+1)^{\frac{3}{2}}\Big[x - \frac{2}{5}(x + 1)\Big] + k\newline[br]= \frac{2}{15}(3x - 2)(x + 1)^{\frac{3}{2}} + k

Or I went horribly wrong, but I don't think I did.

+k

marcusmerehay

+k

Thanks

OP: It seems that your problem wasn't actually doing it, it because you didn't take out the common factors and simplify it at the end.

Reply 16

PeeWeeDan

15

Well the answer I got was far more simple than the answer you guys got... might rearrange to the same, but it seems to differentiate right...

Reply 17

carolcampos

hey guys

I have the same math problem in my text book except they say it must be solved using the substitution method and that u=x+1

I can't figure out how to solve this using substitution. If any of you can help me out I'd really appreciate it :smile:

Reply 18

anonanon123123

14

carolcampos

hey guys

I have the same math problem in my text book except they say it must be solved using the substitution method and that u=x+1

I can't figure out how to solve this using substitution. If any of you can help me out I'd really appreciate it :smile:

u = x + 1 \Rightarrow x = u - 1

So using

u = x + 1, \ \frac{du}{dx} = 1, \ x = u-1

I = \int (u-1) \sqrt{u} \ du = \int u^{\frac{3}{2}} - u^{\frac{1}{2}} \ du

etc

Reply 19

dawnax

To do substitution, let u = sqrt(1+x)

1) square both sides to get u^2=1+x
2) then bring over the 1: you'd get u^2-1=x
3) differentiate both sides: (2u)du=dx
4) sub back into main equation:

Integral x*sqrt(x+1)dx =Integral[ (u^2-1)(u)(2u)du] = Integral [(u^2-1)(2u^2)du]= intergral [2u^4 - 2u^2]du and then you can intergrate simply and substitute u=sqrt(1+x) back into the equation to obtain the

answer:

2/5(1+x)^5/2 - 2/3(1+x)^3/2 + C

:smile:

Integral of x*sqrt(x+1)

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