What is the integral of t(2/7 e^{-(2t)/7}) ?

The integral of t(2/7 e^{-(2t)/7}) is -1/2 e^{-(2t)/7}(2t+7)+C

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integral of t(2/7 e^{-(2t)/7})

\int t (\frac{2}{7} e^{- \frac{2 t}{7}}) d t

- \frac{1}{2} e^{- \frac{2 t}{7}} (2 t + 7) + C

Solution steps

\int t \frac{2}{7} e^{- \frac{2 t}{7}} d t

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{7} \cdot \int t e^{- \frac{2 t}{7}} d t

Apply Integration By Parts

: \frac{1}{2} e^{- \frac{2 t}{7}} t^{2} - \int - \frac{1}{7} e^{- \frac{2 t}{7}} t^{2} d t

= \frac{2}{7} (\frac{1}{2} e^{- \frac{2 t}{7}} t^{2} - \int - \frac{1}{7} e^{- \frac{2 t}{7}} t^{2} d t)

\int - \frac{1}{7} e^{- \frac{2 t}{7}} t^{2} d t = \frac{1}{4} (2 e^{- \frac{2 t}{7}} t^{2} - 49 (- \frac{2 e ^{- \frac{2 t}{7}} t}{7} - e^{- \frac{2 t}{7}}))

= \frac{2}{7} (\frac{1}{2} e^{- \frac{2 t}{7}} t^{2} - \frac{1}{4} (2 e^{- \frac{2 t}{7}} t^{2} - 49 (- \frac{2 e ^{- \frac{2 t}{7}} t}{7} - e^{- \frac{2 t}{7}})))

Simplify

\frac{2}{7} (\frac{1}{2} e^{- \frac{2 t}{7}} t^{2} - \frac{1}{4} (2 e^{- \frac{2 t}{7}} t^{2} - 49 (- \frac{2 e ^{- \frac{2 t}{7}} t}{7} - e^{- \frac{2 t}{7}}))) : - \frac{1}{2} e^{- \frac{2 t}{7}} (2 t + 7)

= - \frac{1}{2} e^{- \frac{2 t}{7}} (2 t + 7)

Add a constant to the solution

= - \frac{1}{2} e^{- \frac{2 t}{7}} (2 t + 7) + C

\int_{- 27}^{0} \frac{1}{9 - x} d x

x \to 0 lim ((\frac{1}{x})^{l n (x)})

x \to - 1 lim (- x^{2} + 5 x - 1)

\int \frac{1}{( x ^{2} + 100 ) ^{\frac{3}{2}}} d x

\frac{( s + 3 )}{s ^{2} + 6 s + 13}

What is the integral of t(2/7 e^{-(2t)/7}) ?
The integral of t(2/7 e^{-(2t)/7}) is -1/2 e^{-(2t)/7}(2t+7)+C