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

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

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integral of x*e^{3x}

\int x \cdot e^{3 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

\frac{d ^{2} y}{d x ^{2}} + 9 y = 10 cos (2 x)

\frac{x ^{3} + 8}{x}

x \to 0 + lim (lo g_{4} (2))

3 y^{^{''}} - 6 y^{^{'}} + 6 y = e^{x} sec (x)

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