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Popular >

integral from ln(2) to ln(4) of (e^{-x})/(sqrt(1-e^{-2x))}

Solution

\int_{l n (2)}^{l n (4)} \frac{e ^{- x}}{1 - e ^{- 2 x}} d x

Solution

arcsin (\frac{15}{4}) - \frac{π}{3}

+1

Decimal

0.27091 \dots

Solution steps

\int_{l n (2)}^{l n (4)} \frac{e ^{- x}}{1 - e ^{- 2 x}} d x

Apply u-substitution

: \int_{\frac{3}{2}}^{\frac{15}{4}} \frac{1}{- u ^{2} + 1} d u

= \int_{\frac{3}{2}}^{\frac{15}{4}} \frac{1}{- u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{- u ^{2} + 1} d u = arcsin (u)

= [arcsin (u)]_{\frac{3}{2}}^{\frac{15}{4}}

Compute the boundaries:

arcsin (\frac{15}{4}) - \frac{π}{3}

= arcsin (\frac{15}{4}) - \frac{π}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the integral from ln(2) to ln(4) of (e^{-x})/(sqrt(1-e^{-2x))} ?
The integral from ln(2) to ln(4) of (e^{-x})/(sqrt(1-e^{-2x))} is arcsin((sqrt(15))/4)-pi/3