What is the integral of y^2e^{3y} ?

The integral of y^2e^{3y} is 1/3 e^{3y}y^2-2/27 (3e^{3y}y-e^{3y})+C

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integral of y^2e^{3y}

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

\frac{1}{3} e^{3 y} y^{2} - \frac{2}{27} (3 e^{3 y} y - e^{3 y}) + C

Solution steps

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

Apply Integration By Parts

= \frac{1}{3} e^{3 y} y^{2} - \int \frac{2}{3} e^{3 y} y d y

\int \frac{2}{3} e^{3 y} y d y = \frac{2}{27} (3 e^{3 y} y - e^{3 y})

= \frac{1}{3} e^{3 y} y^{2} - \frac{2}{27} (3 e^{3 y} y - e^{3 y})

Add a constant to the solution

= \frac{1}{3} e^{3 y} y^{2} - \frac{2}{27} (3 e^{3 y} y - e^{3 y}) + C

\frac{d}{d x} (- 2 cos (x) - x)

\int \frac{1}{t ^{2}} cos (\frac{1}{t} - 1) d t

d er i v a t i v e ln (arctan (x + 1))

\frac{d y}{d x} = \frac{cos ( 8 x )}{e ^{8 y}}

d er i v a t i v e g (x) = (5 - x)^{2} (2 x - 1)^{5}

What is the integral of y^2e^{3y} ?
The integral of y^2e^{3y} is 1/3 e^{3y}y^2-2/27 (3e^{3y}y-e^{3y})+C