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

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

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

\int 2 ln (e^{1 - x}) d x

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

Solution steps

\int 2 ln (e^{1 - x}) d x

Take the constant out:

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

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

Simplify

ln (e^{1 - x}) : 1 - x

= 2 \cdot \int 1 - x d x

Apply u-substitution

: \int - u d u

= 2 \cdot \int - u d u

Take the constant out:

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

= 2 (- \int u d u)

Apply the Power Rule

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

= 2 (- \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 1 - x

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

Simplify

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

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

Add a constant to the solution

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

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