What is the integral of e^{(a+b)x} ?

The integral of e^{(a+b)x} is e^{(a+b)x}x-(e^{x(a+b)}x(a+b)-e^{x(a+b)})/(a+b)+C

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integral of e^{(a+b)x}

\int e^{(a + b) x} d x

e^{(a + b) x} x - \frac{e ^{x (a + b)} x ( a + b ) - e ^{x (a + b)}}{a + b} + C

Solution steps

\int e^{(a + b) x} d x

Apply Integration By Parts

: e^{(a + b) x} x - \int e^{(a + b) x} (a + b) x d x

= e^{(a + b) x} x - \int e^{(a + b) x} (a + b) x d x

\int e^{(a + b) x} (a + b) x d x = \frac{e ^{x (a + b)} x ( a + b ) - e ^{x (a + b)}}{a + b}

= e^{(a + b) x} x - \frac{e ^{x (a + b)} x ( a + b ) - e ^{x (a + b)}}{a + b}

Add a constant to the solution

= e^{(a + b) x} x - \frac{e ^{x (a + b)} x ( a + b ) - e ^{x (a + b)}}{a + b} + C

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What is the integral of e^{(a+b)x} ?
The integral of e^{(a+b)x} is e^{(a+b)x}x-(e^{x(a+b)}x(a+b)-e^{x(a+b)})/(a+b)+C