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

The integral of x/2 (e^x-e^{-x}) is xcosh(x)-sinh(x)+C

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integral of x/2 (e^x-e^{-x})

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

x cosh (x) - sinh (x) + C

Solution steps

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

Take the constant out:

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

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

Apply Integration By Parts

: 2 x cosh (x) - \int 2 cosh (x) d x

= \frac{1}{2} (2 x cosh (x) - \int 2 cosh (x) d x)

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

= \frac{1}{2} (2 x cosh (x) - 2 sinh (x))

Simplify

\frac{1}{2} (2 x cosh (x) - 2 sinh (x)) : x cosh (x) - sinh (x)

= x cosh (x) - sinh (x)

Add a constant to the solution

= x cosh (x) - sinh (x) + C

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What is the integral of x/2 (e^x-e^{-x}) ?
The integral of x/2 (e^x-e^{-x}) is xcosh(x)-sinh(x)+C