Evaluate the definite integral: `int_0^ 6 te^(-t) dt`

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lemjay | High School Teacher | (Level 3) Senior Educator

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`int_0^6te^(-t)dt`

To solve, apply integration by parts. The formula is`int udv=uv-intvdu` .

So let,

`u = t`              and             `dv=e^(-t)dt`

Then, determine du and v.

`du=1*dt`                          `v=int e^(-t)dt`

`du=dt`                            `v=-e^(-t) `

Substitute u,v and du to the formula of integration by parts.

`int te^(-t)dt=t*(-e^(-t)) - int -e^(-t)dt`

                  `= -te^(-t) + inte^(-t)dt`

                  `= -te^(-t) - e^(-t)`

Then, evaluate the limits of integral.

`int_0^6 te^(-t)dt = (-te^(-t) - e^(-t) ) `  `|_0^6`

                `= (-6e^(-6)-e^(-6)) - (0-e^0) = -7e^(-6) +1 = 1-7e^(-6)`

Hence, `int_0^6 te^(-t)dt = 1-7e^(-6)` .

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