Use integration by parts to find the given integral. `I = int55x(ln x)^2 dx = ?`I=(_________________) +c

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lemjay | High School Teacher | (Level 3) Senior Educator

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`I =int 55x(lnx)^2 dx= 55 int x(lnx)^2dx`

To solve using integration by parts, use the formula `int udv =uv -vdu` .

So let,

`u=(lnx)^2`                          and                     `dv= xdx`

Then to determine du, differentiate u. And, take the integral of dv to get v.

`du=2lnx * 1/x dx`                                             `v =int xdx`  

`du=(2lnx)/x dx`                                                `v= x^2/2`        

Plug-in u, v, and dv to the formula.

`55 int x(lnx)^2dx`

`=55[ (lnx)^2*x^2/2 - int x^2/2 *(2lnx)/xdx]`

`=55[(x^2(lnx)^2)/2 - int xlnx dx]`

And to evaluate `intxlnxdx` , apply integration by parts again.

So let,

`u= lnx`                          and           `dv=xdx`

Differentiate u and take the integral of dv again.

`du=1/x dx`                                         `v=int xdx = x^2/2`

And substitute u, v and du to the formula of integration by parts.

`=55[(x^2(lnx)^2)/2- ( ln x * x^2/2 - int x^2/2*1/xdx)]`

`= 55((x^2(lnx)^2)/2 - (x^2lnx)/2 + int x/2 dx)`

`=55((x^2(lnx)^2)/2-(x^2lnx)/2+1/2intxdx)`

`=55[(x^2ln(x)^2)/2-(x^2lnx)/2+1/2*x^2/2]`

`=55((x^2ln(x)^2)/2-(x^2lnx)/2+x^2/4)`

Factor out the GCF.

`=(55x^2)/2 ((lnx)^2-lnx + 1/2)`

Since the given in the problem is indefinite integral, add C.

Hence, `I =(55x^2)/2 ((lnx)^2-lnx + 1/2)+C ` .

 

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