What is the integral of e^{2x}(x^2+3x-1) ?

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

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integral of e^{2x}(x^2+3x-1)

\int e^{2 x} (x^{2} + 3 x - 1) d x

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

Solution steps

\int e^{2 x} (x^{2} + 3 x - 1) d x

Apply Integration By Parts

= \frac{1}{2} e^{2 x} (x^{2} + 3 x - 1) - \int e^{2 x} x + \frac{3}{2} e^{2 x} d x

\int e^{2 x} x + \frac{3}{2} e^{2 x} d x = \frac{1}{2} e^{2 x} x + \frac{1}{2} e^{2 x}

= \frac{1}{2} e^{2 x} (x^{2} + 3 x - 1) - (\frac{1}{2} e^{2 x} x + \frac{1}{2} e^{2 x})

Simplify

\frac{1}{2} e^{2 x} (x^{2} + 3 x - 1) - (\frac{1}{2} e^{2 x} x + \frac{1}{2} e^{2 x}) : \frac{1}{2} e^{2 x} x^{2} + e^{2 x} x - e^{2 x}

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

Add a constant to the solution

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

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