What is the integral of (x+7)e^{2x+3} ?

The integral of (x+7)e^{2x+3} is 1/2 e^{2x+3}x+13/4 e^{2x+3}+C

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integral of (x+7)e^{2x+3}

\int (x + 7) e^{2 x + 3} d x

\frac{1}{2} e^{2 x + 3} x + \frac{13}{4} e^{2 x + 3} + C

Solution steps

\int (x + 7) e^{2 x + 3} d x

Apply Integration By Parts

: \frac{1}{2} e^{2 x + 3} (x + 7) - \int \frac{1}{2} e^{2 x + 3} d x

= \frac{1}{2} e^{2 x + 3} (x + 7) - \int \frac{1}{2} e^{2 x + 3} d x

\int \frac{1}{2} e^{2 x + 3} d x = \frac{1}{4} e^{2 x + 3}

= \frac{1}{2} e^{2 x + 3} (x + 7) - \frac{1}{4} e^{2 x + 3}

Simplify

\frac{1}{2} e^{2 x + 3} (x + 7) - \frac{1}{4} e^{2 x + 3} : \frac{1}{2} e^{2 x + 3} x + \frac{13}{4} e^{2 x + 3}

= \frac{1}{2} e^{2 x + 3} x + \frac{13}{4} e^{2 x + 3}

Add a constant to the solution

= \frac{1}{2} e^{2 x + 3} x + \frac{13}{4} e^{2 x + 3} + C

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x \to - 7 lim (\frac{x ^{2} - 5}{7 - x})

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\frac{d}{d x} ((1 + 3 x)^{\frac{1}{2 x}})

What is the integral of (x+7)e^{2x+3} ?
The integral of (x+7)e^{2x+3} is 1/2 e^{2x+3}x+13/4 e^{2x+3}+C