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

The integral of y^2e^{2y} is 1/2 e^{2y}y^2-1/4 (e^{2y}*2y-e^{2y})+C

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integral of y^2e^{2y}

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

\frac{1}{2} e^{2 y} y^{2} - \frac{1}{4} (e^{2 y} \cdot 2 y - e^{2 y}) + C

Solution steps

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

Apply Integration By Parts

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

\int e^{2 y} y d y = \frac{1}{4} (e^{2 y} \cdot 2 y - e^{2 y})

= \frac{1}{2} e^{2 y} y^{2} - \frac{1}{4} (e^{2 y} \cdot 2 y - e^{2 y})

Add a constant to the solution

= \frac{1}{2} e^{2 y} y^{2} - \frac{1}{4} (e^{2 y} \cdot 2 y - e^{2 y}) + C

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t^{2} \frac{d x}{d t} = \frac{1}{x ^{2}}, x (4) = 9

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What is the integral of y^2e^{2y} ?
The integral of y^2e^{2y} is 1/2 e^{2y}y^2-1/4 (e^{2y}*2y-e^{2y})+C