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

The integral from 4 to x^2 of e^{-sqrt(t)} is 2(-xe^{-x}-e^{-x}+3/(e^2))

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

\int_{4}^{x^{2}} e^{- t} d t

2 (- x e^{- x} - e^{- x} + \frac{3}{e ^{2}})

Solution steps

\int_{4}^{x^{2}} e^{- t} d t

Apply u-substitution

: \int_{- 2}^{- x^{2}} 2 e^{u} u d u

= \int_{- 2}^{- x^{2}} 2 e^{u} u d u

Take the constant out:

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

= 2 \cdot \int_{- 2}^{- x^{2}} e^{u} u d u

Apply Integration By Parts

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

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

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

= 2 [e^{u} u - e^{u}]_{- 2}^{- x^{2}}

Apply radical rule:

n a^{n} = a,

assuming

a \geq 0

= 2 [e^{u} u - e^{u}]_{- 2}^{- x}

Compute the boundaries:

- x e^{- x} - e^{- x} + \frac{3}{e ^{2}}

= 2 (- x e^{- x} - e^{- x} + \frac{3}{e ^{2}})

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What is the integral from 4 to x^2 of e^{-sqrt(t)} ?
The integral from 4 to x^2 of e^{-sqrt(t)} is 2(-xe^{-x}-e^{-x}+3/(e^2))