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

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

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

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

e^{3} (e^{x} x - 3 e^{x}) + C

Solution steps

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

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

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

Take the constant out:

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

= e^{3} \cdot \int (x - 2) e^{x} d x

Apply Integration By Parts

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

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

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

= e^{3} (e^{x} (x - 2) - e^{x})

Expand

e^{x} (x - 2) - e^{x} : e^{x} x - 3 e^{x}

= e^{3} (e^{x} x - 3 e^{x})

Add a constant to the solution

= e^{3} (e^{x} x - 3 e^{x}) + C

\frac{\partial}{\partial x} (4 x^{2} - y^{2} + 2 y)

y = \frac{x}{3}, y = x - 2

\frac{d}{d x} (2 x^{2} + 4 x + 5)

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

\int \frac{8}{x ^{2} + 1} d x

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