What is the integral of e^{-2y}(1+e^{-2y})^{5/2} ?

The integral of e^{-2y}(1+e^{-2y})^{5/2} is -1/7 (1+e^{-2y})^{7/2}+C

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integral of e^{-2y}(1+e^{-2y})^{5/2}

\int e^{- 2 y} (1 + e^{- 2 y})^{\frac{5}{2}} d y

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

Solution steps

\int e^{- 2 y} (1 + e^{- 2 y})^{\frac{5}{2}} d y

Apply u-substitution

: \int - \frac{u ^{\frac{5}{2}}}{2} d u

= \int - \frac{u ^{\frac{5}{2}}}{2} d u

Take the constant out:

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

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

Apply the Power Rule

: \frac{2}{7} u^{\frac{7}{2}}

= - \frac{1}{2} \cdot \frac{2}{7} u^{\frac{7}{2}}

Substitute back

u = 1 + e^{- 2 y}

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

Simplify

- \frac{1}{2} \cdot \frac{2}{7} (1 + e^{- 2 y})^{\frac{7}{2}} : - \frac{1}{7} (1 + e^{- 2 y})^{\frac{7}{2}}

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

Add a constant to the solution

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

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What is the integral of e^{-2y}(1+e^{-2y})^{5/2} ?
The integral of e^{-2y}(1+e^{-2y})^{5/2} is -1/7 (1+e^{-2y})^{7/2}+C