What is the integral of te^{-10t} ?

The integral of te^{-10t} is 1/100 (-10e^{-10t}t-e^{-10t})+C

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integral of te^{-10t}

\int t e^{- 10 t} d t

\frac{1}{100} (- 10 e^{- 10 t} t - e^{- 10 t}) + C

Solution steps

\int t e^{- 10 t} d t

Apply u-substitution

: \int \frac{e ^{u} u}{100} d u

= \int \frac{e ^{u} u}{100} d u

Take the constant out:

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

= \frac{1}{100} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= \frac{1}{100} (e^{u} u - \int e^{u} d u)

\int e^{u} d u = e^{u}

= \frac{1}{100} (e^{u} u - e^{u})

Substitute back

u = - 10 t

= \frac{1}{100} (e^{- 10 t} (- 10 t) - e^{- 10 t})

Simplify

= \frac{1}{100} (- 10 e^{- 10 t} t - e^{- 10 t})

Add a constant to the solution

= \frac{1}{100} (- 10 e^{- 10 t} t - e^{- 10 t}) + C

\int \frac{cos ( x )}{2 - cos ( x )} d x

\frac{d}{d y} (\frac{x ^{2}}{y})

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

\frac{d}{d y} (e^{x} tan (y))

\frac{d x}{d t} = t - x

What is the integral of te^{-10t} ?
The integral of te^{-10t} is 1/100 (-10e^{-10t}t-e^{-10t})+C