What is the integral of 3xy^2e^y ?

The integral of 3xy^2e^y is 3x(y^2e^y-2(e^yy-e^y))+C

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integral of 3xy^2e^y

\int 3 x y^{2} e^{y} d y

3 x (y^{2} e^{y} - 2 (e^{y} y - e^{y})) + C

Solution steps

\int 3 x y^{2} e^{y} d y

Take the constant out:

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

= 3 x \cdot \int e^{y} y^{2} d y

Apply Integration By Parts

: y^{2} e^{y} - \int 2 y e^{y} d y

= 3 x (y^{2} e^{y} - \int 2 y e^{y} d y)

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

= 3 x (y^{2} e^{y} - 2 (e^{y} y - e^{y}))

Add a constant to the solution

= 3 x (y^{2} e^{y} - 2 (e^{y} y - e^{y})) + C

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What is the integral of 3xy^2e^y ?
The integral of 3xy^2e^y is 3x(y^2e^y-2(e^yy-e^y))+C