How do you integrate $\int x e^{\frac{x}{2}}$ $int xe^(x/2)$ using integration by parts?

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How do you integrate $\int x e^{\frac{x}{2}}$ $int xe^(x/2)$ using integration by parts?

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Answer 1 · 2018-04-20T18:40:26

$\int x e^{\frac{x}{2}} d x = 2 e^{\frac{x}{2}} (x - 2) + C$ $intxe^(x/2)dx=2e^(x/2)(x-2)+C$

Explanation:

Make the following selections:

$u = x$ $u=x$

$d u = d x$ $du=dx$

$d v = e^{\frac{x}{2}} d x$ $dv=e^(x/2)dx$

$v = \int e^{\frac{x}{2}} d x = 2 e^{\frac{x}{2}}$ $v=inte^(x/2)dx=2e^(x/2)$

Then

$u v - \int v d u = 2 x e^{\frac{x}{2}} - 2 \int e^{\frac{x}{2}} d x = 2 x e^{\frac{x}{2}} - 4 e^{\frac{x}{2}} + C$ $uv-intvdu=2xe^(x/2)-2inte^(x/2)dx=2xe^(x/2)-4e^(x/2)+C$

Factoring out the exponential, we obtain

$\int x e^{\frac{x}{2}} d x = 2 e^{\frac{x}{2}} (x - 2) + C$ $intxe^(x/2)dx=2e^(x/2)(x-2)+C$

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How do you integrate $\int x e^{\frac{x}{2}}$ $int xe^(x/2)$ using integration by parts?

1 Answer

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