What is the integral of (5-3x)e^{2x+3} ?

The integral of (5-3x)e^{2x+3} is e^3(-3/2 e^{2x}x+13/4 e^{2x})+C

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integral of (5-3x)e^{2x+3}

\int (5 - 3 x) e^{2 x + 3} d x

e^{3} (- \frac{3}{2} e^{2 x} x + \frac{13}{4} e^{2 x}) + C

Solution steps

\int (5 - 3 x) e^{2 x + 3} d x

(5 - 3 x) e^{2 x + 3} = e^{3} (5 - 3 x) e^{2 x}

= \int e^{3} (5 - 3 x) e^{2 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 (5 - 3 x) e^{2 x} d x

Apply Integration By Parts

= e^{3} (\frac{1}{2} e^{2 x} (5 - 3 x) - \int - \frac{3}{2} e^{2 x} d x)

\int - \frac{3}{2} e^{2 x} d x = - \frac{3}{4} e^{2 x}

= e^{3} (\frac{1}{2} e^{2 x} (5 - 3 x) - (- \frac{3}{4} e^{2 x}))

Simplify

e^{3} (\frac{1}{2} e^{2 x} (5 - 3 x) - (- \frac{3}{4} e^{2 x})) : e^{3} (- \frac{3}{2} e^{2 x} x + \frac{13}{4} e^{2 x})

= e^{3} (- \frac{3}{2} e^{2 x} x + \frac{13}{4} e^{2 x})

Add a constant to the solution

= e^{3} (- \frac{3}{2} e^{2 x} x + \frac{13}{4} e^{2 x}) + C

What is the integral of (5-3x)e^{2x+3} ?
The integral of (5-3x)e^{2x+3} is e^3(-3/2 e^{2x}x+13/4 e^{2x})+C