What is the integral from 1 to 4 of 4e^{sqrt(x)} ?

The integral from 1 to 4 of 4e^{sqrt(x)} is 8e^2

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integral from 1 to 4 of 4e^{sqrt(x)}

\int_{1}^{4} 4 e^{x} d x

8 e^{2}

Decimal

59.11244 \dots

Solution steps

\int_{1}^{4} 4 e^{x} d x

Take the constant out:

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

= 4 \cdot \int_{1}^{4} e^{x} d x

Apply u-substitution

: \int_{1}^{2} e^{u} \cdot 2 u d u

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

Take the constant out:

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

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

Apply Integration By Parts

: [e^{u} u - \int e^{u} d u]_{1}^{2}

= 4 \cdot 2 [e^{u} u - \int e^{u} d u]_{1}^{2}

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

= 4 \cdot 2 [e^{u} u - e^{u}]_{1}^{2}

Simplify

= 8 [e^{u} u - e^{u}]_{1}^{2}

Compute the boundaries:

e^{2}

= 8 e^{2}

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What is the integral from 1 to 4 of 4e^{sqrt(x)} ?
The integral from 1 to 4 of 4e^{sqrt(x)} is 8e^2