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

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

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

\int_{1}^{4} 2 e^{x} d x

4 e^{2}

Decimal Notation

29.55622 \dots

Solution steps

\int_{1}^{4} 2 e^{x} d x

Take the constant out:

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

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

Apply u-substitution

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

= 2 \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

= 2 \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}

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

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

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

Simplify

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

Compute the boundaries:

e^{2}

= 4 e^{2}

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\frac{d y}{d x}, x + y = 1

x = - 2 y^{2} - 4 y + 1, x = y^{2} + 8 y + 10

\frac{\partial}{\partial x} (e^{3 x yz - 8})

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