Integral Of Xe^2x

what is the integral of xe^2x ?

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This should be soled as integrals by parts. In Integrals by parts it says if U and V are fuctions then,

`intUdv = UV-intVdu`

So let us take U= x and V=e^(2x)

Then by differentiation with respect to x;

dU/dx = 1

dU = dx

dV/dx = 2e^(2x)

dV = 2e^(2x) dx

By applying the intergration by parts;

`int(x*2e^(2x)) dx = x*e^(2x)-int(e^(2x)) dx`

`intx*e^(2x) dx =` (1/2)(`x*e^(2x)-int(e^(2x)) dx`)

=` (1/2)(x*e^(2x)-1/2*e^(2x))+C`

So the answer is;

`int(x*e^(2x)) dx ` = ` (1/2)(x*e^(2x)-1/2*e^(2x))+C`

Where C is a contant.

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