What is the integral of (x+5)e^{x+3} ?

The integral of (x+5)e^{x+3} is e^3(e^xx+4e^x)+C

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integral of (x+5)e^{x+3}

\int (x + 5) e^{x + 3} d x

e^{3} (e^{x} x + 4 e^{x}) + C

Solution steps

\int (x + 5) e^{x + 3} d x

(x + 5) e^{x + 3} = e^{3} (x + 5) e^{x}

= \int e^{3} (x + 5) 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 + 5) e^{x} d x

Apply Integration By Parts

= e^{3} (e^{x} (x + 5) - \int e^{x} d x)

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

= e^{3} (e^{x} (x + 5) - e^{x})

Expand

e^{x} (x + 5) - e^{x} : e^{x} x + 4 e^{x}

= e^{3} (e^{x} x + 4 e^{x})

Add a constant to the solution

= e^{3} (e^{x} x + 4 e^{x}) + C

What is the integral of (x+5)e^{x+3} ?
The integral of (x+5)e^{x+3} is e^3(e^xx+4e^x)+C