What is the integral from 0 to 0.5 of xe^{2sqrt(x)} ?

The integral from 0 to 0.5 of xe^{2sqrt(x)} is 0.39713…

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integral from 0 to 0.5 of xe^{2sqrt(x)}

\int_{0}^{0.5} x e^{2 x} d x

0.39713 \dots

Solution steps

\int_{0}^{0.5} x e^{2 x} d x

Apply Integration By Parts

: [x (e^{2 x} x - 0.5 e^{2 x}) - \int e^{2 x} x - 0.5 e^{2 x} d x]_{0}^{0.5}

= [x (e^{2 x} x - 0.5 e^{2 x}) - \int e^{2 x} x - 0.5 e^{2 x} d x]_{0}^{0.5}

\int e^{2 x} x - 0.5 e^{2 x} d x = e^{2 x} x - 1.5 e^{2 x} x + 0.75 e^{2 x}

= [x (e^{2 x} x - 0.5 e^{2 x}) - (e^{2 x} x - 1.5 e^{2 x} x + 0.75 e^{2 x})]_{0}^{0.5}

Expand

x (e^{2 x} x - 0.5 e^{2 x}) - (e^{2 x} x - 1.5 e^{2 x} x + 0.75 e^{2 x}) : e^{2 x} x x - 1.5 e^{2 x} x + 1.5 e^{2 x} x - 0.75 e^{2 x}

= [e^{2 x} x x - 1.5 e^{2 x} x + 1.5 e^{2 x} x - 0.75 e^{2 x}]_{0}^{0.5}

Compute the boundaries:

0.39713 \dots

= 0.39713 \dots

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What is the integral from 0 to 0.5 of xe^{2sqrt(x)} ?
The integral from 0 to 0.5 of xe^{2sqrt(x)} is 0.39713…