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

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

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

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

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

Solution steps

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

= \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

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