What is the integral from 0 to 1 of 2pix(e^x-e^{-x}) ?

The integral from 0 to 1 of 2pix(e^x-e^{-x}) is (4pi)/e

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integral from 0 to 1 of 2pix(e^x-e^{-x})

\int_{0}^{1} 2 π x (e^{x} - e^{- x}) d x

\frac{4 π}{e}

Decimal

4.62290 \dots

Solution steps

\int_{0}^{1} 2 π x (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 π \cdot \int_{0}^{1} x (e^{x} - e^{- x}) d x

Apply Integration By Parts

= 2 π [2 x cosh (x) - \int 2 cosh (x) d x]_{0}^{1}

\int 2 cosh (x) d x = 2 sinh (x)

= 2 π [2 x cosh (x) - 2 sinh (x)]_{0}^{1}

Compute the boundaries:

\frac{2}{e}

= 2 π \frac{2}{e}

Simplify

= \frac{4 π}{e}

What is the integral from 0 to 1 of 2pix(e^x-e^{-x}) ?
The integral from 0 to 1 of 2pix(e^x-e^{-x}) is (4pi)/e