Solved: "Evaluate the integral. ∫te−4tdt"

Lennie Carroll

Answered question

2021-11-08

Evaluate the integral.

\int t e^{- 4 t} d t

Anonym

Skilled2021-11-09Added 108 answers

Step 1
Given data:
The given integral is:

\int t e^{- 4 t} d t

.
The integration of the expression can be solved by parts.

\int t e^{- 4 t} d t = t \int e^{- 4 t} d t - \int (\frac{d t}{d t} \int e^{- 4 t} d t) d t

= t (\frac{e^{- 4 t}}{- 4}) - \int (\frac{e^{- 4 t}}{- 4}) d t

= - \frac{t e^{- 4 t}}{4} + \frac{1}{4} \int e^{- 4 t} d t

Step 2
Further, solve the above-integral.

\int t e^{- 4 t} d t = - \frac{t e^{- 4 t}}{4} + \frac{1}{4} \int e^{- 4 t} d t

= - \frac{t e^{- 4 t}}{4} + \frac{1}{4} (\frac{e^{- 4 t}}{- 4}) + C

= - \frac{e^{- 4 t}}{4} (t + \frac{1}{4}) + C

Thus, the integral of the expression is

[- e^{- 4 t} (t + \frac{1}{4})] 4 + c

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