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

The integral from 2 to 3 of (x-2)e^{2x} is (5e^6-3e^4)/4-e^6+e^4

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integral from 2 to 3 of (x-2)e^{2x}

\int_{2}^{3} (x - 2) e^{2 x} d x

\frac{5 e ^{6} - 3 e ^{4}}{4} - e^{6} + e^{4}

Solution steps

\int_{2}^{3} (x - 2) e^{2 x} d x

Expand

(x - 2) e^{2 x} : e^{2 x} x - 2 e^{2 x}

= \int_{2}^{3} e^{2 x} x - 2 e^{2 x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{2}^{3} e^{2 x} x d x - \int_{2}^{3} 2 e^{2 x} d x

\int_{2}^{3} e^{2 x} x d x = \frac{5 e ^{6} - 3 e ^{4}}{4}

\int_{2}^{3} 2 e^{2 x} d x = e^{6} - e^{4}

= \frac{5 e ^{6} - 3 e ^{4}}{4} - (e^{6} - e^{4})

Simplify

= \frac{5 e ^{6} - 3 e ^{4}}{4} - e^{6} + e^{4}

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What is the integral from 2 to 3 of (x-2)e^{2x} ?
The integral from 2 to 3 of (x-2)e^{2x} is (5e^6-3e^4)/4-e^6+e^4