What is the integral of (x+1)^2e^{2x} ?

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

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integral of (x+1)^2e^{2x}

\int (x + 1)^{2} e^{2 x} d x

\frac{1}{2} e^{2 x} x^{2} + \frac{1}{2} e^{2 x} x + \frac{1}{4} e^{2 x} + C

Solution steps

\int (x + 1)^{2} e^{2 x} d x

Apply Integration By Parts

: \frac{1}{2} e^{2 x} x^{2} + e^{2 x} x + \frac{1}{2} e^{2 x} - \int e^{2 x} (x + 1) d x

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

\int e^{2 x} (x + 1) d x = \frac{1}{2} e^{2 x} x + \frac{1}{4} e^{2 x}

= \frac{1}{2} e^{2 x} x^{2} + e^{2 x} x + \frac{1}{2} e^{2 x} - (\frac{1}{2} e^{2 x} x + \frac{1}{4} e^{2 x})

Simplify

\frac{1}{2} e^{2 x} x^{2} + e^{2 x} x + \frac{1}{2} e^{2 x} - (\frac{1}{2} e^{2 x} x + \frac{1}{4} e^{2 x}) : \frac{1}{2} e^{2 x} x^{2} + \frac{1}{2} e^{2 x} x + \frac{1}{4} e^{2 x}

= \frac{1}{2} e^{2 x} x^{2} + \frac{1}{2} e^{2 x} x + \frac{1}{4} e^{2 x}

Add a constant to the solution

= \frac{1}{2} e^{2 x} x^{2} + \frac{1}{2} e^{2 x} x + \frac{1}{4} e^{2 x} + C

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x \to 0 lim (\frac{( x + h ) ^{3} - x}{h})

\frac{d y}{d x} + y = e^{3 x}

\frac{\partial}{\partial x} (2 x^{3} - 54 x - 2 y^{2} - 12 y)

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