What is the integral of xe^{8x} ?

The integral of xe^{8x} is 1/64 (e^{8x}*8x-e^{8x})+C

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integral of xe^{8x}

\int x e^{8 x} d x

\frac{1}{64} (e^{8 x} \cdot 8 x - e^{8 x}) + C

Solution steps

\int x e^{8 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{64} d u

= \int \frac{e ^{u} u}{64} d u

Take the constant out:

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

= \frac{1}{64} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= \frac{1}{64} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{64} (e^{u} u - e^{u})

Substitute back

u = 8 x

= \frac{1}{64} (e^{8 x} \cdot 8 x - e^{8 x})

Add a constant to the solution

= \frac{1}{64} (e^{8 x} \cdot 8 x - e^{8 x}) + C

\frac{d}{d x} (8^{\frac{1}{3}})

x^{\frac{2}{3}}

x \to e lim (x (e^{x} + \frac{1}{x}))

x \to 0 lim (\frac{( 4 ^{x} - 2 ^{x} )}{x})

x \to 0 lim ((1 - 3 x)^{\frac{1}{x}})

What is the integral of xe^{8x} ?
The integral of xe^{8x} is 1/64 (e^{8x}*8x-e^{8x})+C