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

The integral of (e^{x/2}-e^{-x/2}) is 4cosh(x/2)+C

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

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

4 cosh (\frac{x}{2}) + C

Solution steps

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

Apply u-substitution

: \int (e^{u} - e^{- u}) \cdot 2 d u

= \int (e^{u} - e^{- u}) \cdot 2 d u

Take the constant out:

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

= 2 \cdot \int e^{u} - e^{- u} d u

Rewrite using trig identities

= 2 \cdot \int 2 sinh (u) d u

Take the constant out:

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

= 2 \cdot 2 \cdot \int sinh (u) d u

Use the common integral:

\int sinh (u) d u = cosh (u)

= 2 \cdot 2 cosh (u)

Substitute back

u = \frac{x}{2}

= 2 \cdot 2 cosh (\frac{x}{2})

Simplify

= 4 cosh (\frac{x}{2})

Add a constant to the solution

= 4 cosh (\frac{x}{2}) + C

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127 q + 1406

\int_{0}^{7} (7 - x) d x

f (x) = x^{x}

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