What is the integral of (1+e^{-at})^{3/2}e^{-at} ?

The integral of (1+e^{-at})^{3/2}e^{-at} is -(2(1+e^{-at})^{5/2})/(5a)+C

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integral of (1+e^{-at})^{3/2}e^{-at}

\int (1 + e^{- a t})^{\frac{3}{2}} e^{- a t} d t

- \frac{2 ( 1 + e ^{- a t} ) ^{\frac{5}{2}}}{5 a} + C

Solution steps

\int (1 + e^{- a t})^{\frac{3}{2}} e^{- a t} d t

Apply u-substitution

: \int - \frac{u ^{\frac{3}{2}}}{a} d u

= \int - \frac{u ^{\frac{3}{2}}}{a} d u

Take the constant out:

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

= - \frac{1}{a} \cdot \int u^{\frac{3}{2}} d u

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= - \frac{1}{a} \cdot \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = 1 + e^{- a t}

= - \frac{1}{a} \cdot \frac{2}{5} (1 + e^{- a t})^{\frac{5}{2}}

Simplify

- \frac{1}{a} \cdot \frac{2}{5} (1 + e^{- a t})^{\frac{5}{2}} : - \frac{2 ( 1 + e ^{- a t} ) ^{\frac{5}{2}}}{5 a}

= - \frac{2 ( 1 + e ^{- a t} ) ^{\frac{5}{2}}}{5 a}

Add a constant to the solution

= - \frac{2 ( 1 + e ^{- a t} ) ^{\frac{5}{2}}}{5 a} + C

3 e^{x} + \frac{4}{3 x}

y = (\frac{1}{2 x + 1})^{2}

x \to a lim ((1 + \frac{1}{x})^{x})

\frac{d}{d x} (\frac{1}{10 - x})

\int (y e^{x y}) d x

What is the integral of (1+e^{-at})^{3/2}e^{-at} ?
The integral of (1+e^{-at})^{3/2}e^{-at} is -(2(1+e^{-at})^{5/2})/(5a)+C