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

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

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

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

\frac{e ^{4} - 1}{2 e ^{2}}

Decimal

3.62686 \dots

Solution steps

\int_{0}^{2} \frac{1}{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_{0}^{2} e^{x} + e^{- x} d x

Rewrite using trig identities

= \frac{1}{2} \cdot \int_{0}^{2} 2 cosh (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 2 \cdot \int_{0}^{2} cosh (x) d x

Use the common integral:

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

= \frac{1}{2} \cdot 2 [sinh (x)]_{0}^{2}

Simplify

\frac{1}{2} \cdot 2 [sinh (x)]_{0}^{2} : [sinh (x)]_{0}^{2}

= [sinh (x)]_{0}^{2}

Compute the boundaries:

\frac{e ^{4} - 1}{2 e ^{2}}

= \frac{e ^{4} - 1}{2 e ^{2}}

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