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How do you integrate #xlnx#?

Answer 1

Emily Ammerman

It is

#int xlnxdx=int (x^2/2)'lnxdx=x^2/2lnx-intx^2/2lnx'dx= x^2/2lnx-intx^2/21/xdx=x^2/2lnx-intx/2dx=x^2/2lnx-x^2/4+c#

Finally #int xlnxdx=x^2/2*lnx-x^2/4+c#

Integration by portions was employed.

#intf'(x)*g(x)dx=f(x)g(x)-intf(x)*g'(x)dx#

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Answer 2

Cameron Alexander

To integrate ( x \ln(x) ), you can use integration by parts. Integration by parts is given by the formula:

[ \int u , dv = uv - \int v , du ]

For ( x \ln(x) ), you can let ( u = \ln(x) ) and ( dv = x , dx ). Then, you find ( du ) and ( v ), and apply the integration by parts formula.

[ u = \ln(x), , dv = x , dx ] [ du = \frac{1}{x} , dx, , v = \frac{x^2}{2} ]

Now, applying the integration by parts formula:

[ \int x \ln(x) , dx = \frac{x^2 \ln(x)}{2} - \int \frac{x^2}{2} \frac{1}{x} , dx ]

Simplify the integral:

[ = \frac{x^2 \ln(x)}{2} - \frac{1}{2} \int x , dx ]

[ = \frac{x^2 \ln(x)}{2} - \frac{x^2}{4} + C ]

Where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.