What is the integral of 2sqrt(1-e^{-x)}e^{-x} ?

The integral of 2sqrt(1-e^{-x)}e^{-x} is 4/3 (1-e^{-x})^{3/2}+C

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integral of 2sqrt(1-e^{-x)}e^{-x}

\int 2 1 - e^{- x} e^{- x} d x

\frac{4}{3} (1 - e^{- x})^{\frac{3}{2}} + C

Solution steps

\int 2 1 - e^{- x} e^{- x} d x

Take the constant out:

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

= 2 \cdot \int 1 - e^{- x} e^{- x} d x

Apply u-substitution

: \int u d u

= 2 \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= 2 \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 - e^{- x}

= 2 \cdot \frac{2}{3} (1 - e^{- x})^{\frac{3}{2}}

Simplify

2 \cdot \frac{2}{3} (1 - e^{- x})^{\frac{3}{2}} : \frac{4}{3} (1 - e^{- x})^{\frac{3}{2}}

= \frac{4}{3} (1 - e^{- x})^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{3} (1 - e^{- x})^{\frac{3}{2}} + C

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What is the integral of 2sqrt(1-e^{-x)}e^{-x} ?
The integral of 2sqrt(1-e^{-x)}e^{-x} is 4/3 (1-e^{-x})^{3/2}+C