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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

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

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

Substitute back

u = 1 + e^{- π t}

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

Simplify

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

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

Add a constant to the solution

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

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