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

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

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

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

e^{5} (e^{3 x} x + \frac{4}{3} e^{3 x}) + C

Solution steps

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

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

\int e^{3 x} d x = \frac{1}{3} e^{3 x}

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

Add similar elements:

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

= e^{5} (e^{3 x} x + \frac{4}{3} e^{3 x})

Add a constant to the solution

= e^{5} (e^{3 x} x + \frac{4}{3} e^{3 x}) + C

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