What is the integral from 0 to 4 of xe^{sqrt(x)} ?

The integral from 0 to 4 of xe^{sqrt(x)} is 2(2e^2+6)

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integral from 0 to 4 of xe^{sqrt(x)}

\int_{0}^{4} x e^{x} d x

2 (2 e^{2} + 6)

Decimal

41.55622 \dots

Solution steps

\int_{0}^{4} x e^{x} d x

Apply u-substitution

: \int_{0}^{2} 2 e^{u} u^{3} d u

= \int_{0}^{2} 2 e^{u} u^{3} d u

Take the constant out:

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

= 2 \cdot \int_{0}^{2} e^{u} u^{3} d u

Apply Integration By Parts

: [u^{3} e^{u} - \int 3 u^{2} e^{u} d u]_{0}^{2}

= 2 [u^{3} e^{u} - \int 3 u^{2} e^{u} d u]_{0}^{2}

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= 2 [u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))]_{0}^{2}

Compute the boundaries:

2 e^{2} + 6

= 2 (2 e^{2} + 6)

\frac{d}{d t} (\frac{t ^{3}}{6} + \frac{t ^{2}}{4} + \frac{1}{2} t)

\int \frac{sec ( x ) tan ( x )}{sec ( x ) - 7} d x

\frac{d}{d θ} (- 2 sin (θ))

e^{t} sin (t)

\frac{\partial}{\partial x} (4 x^{- \frac{1}{8}})

What is the integral from 0 to 4 of xe^{sqrt(x)} ?
The integral from 0 to 4 of xe^{sqrt(x)} is 2(2e^2+6)