What is the integral from 0 to 2 of pie^{2x} ?

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

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integral from 0 to 2 of pie^{2x}

\int_{0}^{2} π e^{2 x} d x

\frac{π ( e ^{4} - 1 )}{2}

Decimal

84.19177 \dots

Solution steps

\int_{0}^{2} π e^{2 x} d x

Take the constant out:

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

= π \cdot \int_{0}^{2} e^{2 x} d x

Apply u-substitution

: \int_{0}^{4} e^{u} \frac{1}{2} d u

= π \cdot \int_{0}^{4} e^{u} \frac{1}{2} d u

Take the constant out:

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

= π \frac{1}{2} \cdot \int_{0}^{4} e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= π \frac{1}{2} [e^{u}]_{0}^{4}

Simplify

π \frac{1}{2} [e^{u}]_{0}^{4} : \frac{π}{2} [e^{u}]_{0}^{4}

= \frac{π}{2} [e^{u}]_{0}^{4}

Compute the boundaries:

e^{4} - 1

= \frac{π}{2} (e^{4} - 1)

Simplify

= \frac{π ( e ^{4} - 1 )}{2}

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