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

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

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

= \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))

d er i v a t i v e 2 4 9 - x^{2}

\int \frac{cos ( θ )}{2 + sin ^{2} ( θ )} d θ

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

\frac{\partial}{\partial x} ((41 - x^{2} - y^{2})^{\frac{1}{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