What is the integral of xlnx?

i'll do it by 2 ways

the first one if u know what integration by parts is:

(int=integral)

int[udv]=uv-int[vdu]

let u=lnx =>du=dx/x

& let dv=xdx => v=(x^2)/2

so we'll have

int[xlnx]=((x^2)/2) lnx -1/2int[xdx]

=((x^2)/2)lnx -((x^2)/4)+ Constante

the second method is

try derivating (x^2)lnx u'll get [(x^2)lnx]'=2xlnx +x

then integrate int{[(x^2)lnx]dx}=(x^2)lnx

&int[x]=(x^2)/2

so u'll get (x^2)lnx=2int[xlnxdx]+(x^2)/2

so int[xlnxdx]=((x^2)/2)lnx-(x^2)/4+ C

Integral Of Xlnx

You need to use integration by parts so that (cap. S stands for ingegral)

Su*dv = uv - Sv*du

So for Sxln(x) you need to choose your u and dv, then use those to solve for du and v. ln(x) is not easy to integrate, but it is easy to differentiate, so choose it as your "u", therefore xdx would be "dv"

u = ln(x); dv = xdx

du = (1/x)dx; v = 1/2*x^2

So now you just plug it into the formula above

Sudv = uv - Svdu

= ln(x)*(x^2/2) - S(x^2/2)*(dx/x) --- pull out the 1/2, and simplify x^2/x

= ln(x)*(x^2/2) - 1/2*[Sxdx]

= 1/2*(x^2ln(x)) - 1/2*[1/2*x^2]

= 1/2*(x^2ln(x)) - 1/4(x^2) + C

You can keep simplifying from here if you want, but I'd probably leave it like this.

Use integration by parts.

Let u=ln x, and let dv=xdx.

Then,

du=dx/x, and

v=(x^2)/2

By parts

Integral(xlnxdx) = uv-Integral(vdu) = (x^2)*lnx/2 - Integral [((x^2)/2)*(1/x)dx]

...=(x^2)*lnx/2 - Integral(xdx/2)

...=(x^2)*lnx/2 - (x^2)/4 + C

...=((x^2)/2)*(lnx - 1/2) + C

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you are right, the answer is ln(u) + C = ln [ln(x) ] + C substitution is very simple. The given integral is dx/ (x ln x) = (1/ln x)( dx/x ) substitute ln(x) = u ==> dx/x = du , so the integral becomes (1/u)(du) = du/u

RE:

What is the integral of xlnx?

Maths Problem - if you could help with this integral, it would be much appreciated.

I've put a beautifully formatted PDF here. Take a look. It looks as it would in a textbook. Each step is clearly annotated.

http://www.tomsmath.com/step-by-step-instructions-...

take ln x as u and (x)squared dx as dv

du= dx/x

v= (x)squared/2

then use the formula and integrate

your final answer should be something like

(x)squared ln x/2 - (x)squared/2 +c.

(x^2*ln*(x))/2 - (x^2)/4 + c