What is the integral from 0 to 1 of sqrt(1-e^{-x)} ?

The integral from 0 to 1 of sqrt(1-e^{-x)} is ln(sqrt(-1/e+1)+1)-ln(-sqrt(-1/e+1)+1)-2sqrt(-1/e+1)

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integral from 0 to 1 of sqrt(1-e^{-x)}

\int_{0}^{1} 1 - e^{- x} d x

ln (- \frac{1}{e} + 1 + 1) - ln (- - \frac{1}{e} + 1 + 1) - 2 - \frac{1}{e} + 1

Decimal

0.57995 \dots

Solution steps

\int_{0}^{1} 1 - e^{- x} d x

Apply u-substitution

= \int_{1}^{\frac{1}{e}} - \frac{1 - u}{u} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{\frac{1}{e}}^{1} - \frac{1 - u}{u} d u

Take the constant out:

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

= - (- \int_{\frac{1}{e}}^{1} \frac{1 - u}{u} d u)

Apply u-substitution

= - (- \int_{1 - \frac{1}{e}}^{0} - \frac{2 v ^{2}}{- v ^{2} + 1} d v)

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - (- (- \int_{0}^{1 - \frac{1}{e}} - \frac{2 v ^{2}}{- v ^{2} + 1} d v))

Take the constant out:

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

= - (- (- (- 2 \cdot \int_{0}^{1 - \frac{1}{e}} \frac{v ^{2}}{- v ^{2} + 1} d v)))

\frac{v ^{2}}{- v ^{2} + 1} = \frac{1}{- v ^{2} + 1} - 1

= - (- (- (- 2 \cdot \int_{0}^{1 - \frac{1}{e}} \frac{1}{- v ^{2} + 1} - 1 d v)))

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- (- (- 2 (\int_{0}^{1 - \frac{1}{e}} \frac{1}{- v ^{2} + 1} d v - \int_{0}^{1 - \frac{1}{e}} 1 d v))))

\int_{0}^{1 - \frac{1}{e}} \frac{1}{- v ^{2} + 1} d v = \frac{1}{2} (ln (- \frac{1}{e} + 1 + 1) - ln (- - \frac{1}{e} + 1 + 1))

\int_{0}^{1 - \frac{1}{e}} 1 d v = - \frac{1}{e} + 1

= - (- (- (- 2 (\frac{1}{2} (ln (- \frac{1}{e} + 1 + 1) - ln (- - \frac{1}{e} + 1 + 1)) - - \frac{1}{e} + 1))))

Simplify

- (- (- (- 2 (\frac{1}{2} (ln (- \frac{1}{e} + 1 + 1) - ln (- - \frac{1}{e} + 1 + 1)) - - \frac{1}{e} + 1)))) : ln (- \frac{1}{e} + 1 + 1) - ln (- - \frac{1}{e} + 1 + 1) - 2 - \frac{1}{e} + 1

= ln (- \frac{1}{e} + 1 + 1) - ln (- - \frac{1}{e} + 1 + 1) - 2 - \frac{1}{e} + 1

What is the integral from 0 to 1 of sqrt(1-e^{-x)} ?
The integral from 0 to 1 of sqrt(1-e^{-x)} is ln(sqrt(-1/e+1)+1)-ln(-sqrt(-1/e+1)+1)-2sqrt(-1/e+1)