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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{e ^{u}}{2} d u

= \int - \frac{e ^{u}}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Use the common integral:

\int e^{u} d u = e^{u}

= - \frac{1}{2} e^{u}

Substitute back

u = - 2 (x^{2} + 3 x + 1)

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

Add a constant to the solution

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

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