What is the integral of x*e^{3*x} ?

The integral of x*e^{3*x} is 1/9 (e^{3x}*3x-e^{3x})+C

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integral of xe^{3x}

\int x \cdot e^{3 \cdot x} d x

\frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x}) + C

Solution steps

\int x e^{3 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{9} d u

= \int \frac{e ^{u} u}{9} d u

Take the constant out:

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

= \frac{1}{9} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= \frac{1}{9} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{9} (e^{u} u - e^{u})

Substitute back

u = 3 x

= \frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x})

Add a constant to the solution

= \frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x}) + C

x \to - 3 lim (5 (3 x^{2} - 24 x + 3))

\int_{0}^{4} π (2 - x)^{2} d x

f (x) = e^{2 x} \cdot 2

\frac{d}{d x} (2 e^{\frac{x}{2}})

\frac{d x}{d y} = \frac{y}{x ^{2}}

What is the integral of x*e^{3*x} ?
The integral of x*e^{3*x} is 1/9 (e^{3x}*3x-e^{3x})+C