# use integration by parts to find the definite integral int^(1_0) xe^(0.2x) dx

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embizze | Certified Educator

Evaluate `int_0^1xe^(.2x)dx` :

Since `e^(.2x)` is more complicated, and we know its antiderivative, use the following substitutions:

`u=x` `dv=e^(.2x)`

`du=dx` `v=inte^(.2x)dx=5e^(.2x)`

Then `intudv=uv-intvdu` so:

`int_0^1xe^(.2x)dx=[x*5e^(.2x)-int5e^(.2x)dx]_0^1`

`=[5xe^(.2x)-5(5e^(.2x))]_0^1`

`=[5xe^(.2x)-25e^(.2x)]_0^1`

`=(5e^(.2)-25e^(.2))-(0-25)`

`=-20e^(.2)+25`

`~~.572`

The graph :

As you see, the area under the graph appears to be a little more than 1/2.