What is the integral of 2xe^{xy} ?

The integral of 2xe^{xy} is 2(e^{yx} 1/y x-e^{yx} 1/(y^2))+C

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integral of 2xe^{xy}

\int 2 x e^{x y} d x

2 (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}}) + C

Solution steps

\int 2 x e^{x y} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int e^{y x} x d x

Apply Integration By Parts

: e^{y x} \frac{1}{y} x - \int e^{y x} \frac{1}{y} d x

= 2 (e^{y x} \frac{1}{y} x - \int e^{y x} \frac{1}{y} d x)

\int e^{y x} \frac{1}{y} d x = e^{y x} \frac{1}{y ^{2}}

= 2 (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}})

Add a constant to the solution

= 2 (e^{y x} \frac{1}{y} x - e^{y x} \frac{1}{y ^{2}}) + C

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What is the integral of 2xe^{xy} ?
The integral of 2xe^{xy} is 2(e^{yx} 1/y x-e^{yx} 1/(y^2))+C