What is the integral of 2/3 e^{3x}x ?

The integral of 2/3 e^{3x}x is 2/27 (3e^{3x}x-e^{3x})+C

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integral of 2/3 e^{3x}x

\int \frac{2}{3} e^{3 x} x d x

\frac{2}{27} (3 e^{3 x} x - e^{3 x}) + C

Solution steps

\int \frac{2}{3} e^{3 x} x d x

Take the constant out:

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

= \frac{2}{3} \cdot \int e^{3 x} x d x

Apply u-substitution

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

= \frac{2}{3} \cdot \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{2}{3} \cdot \frac{1}{9} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= \frac{2}{3} \cdot \frac{1}{9} (e^{u} u - \int e^{u} d u)

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

= \frac{2}{3} \cdot \frac{1}{9} (e^{u} u - e^{u})

Substitute back

u = 3 x

= \frac{2}{3} \cdot \frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x})

Simplify

\frac{2}{3} \cdot \frac{1}{9} (e^{3 x} \cdot 3 x - e^{3 x}) : \frac{2}{27} (3 e^{3 x} x - e^{3 x})

= \frac{2}{27} (3 e^{3 x} x - e^{3 x})

Add a constant to the solution

= \frac{2}{27} (3 e^{3 x} x - e^{3 x}) + C

\frac{d}{d x} (x^{4} + 5 x)

\frac{\partial}{\partial y} (x^{4} y - 2 x^{3} y^{2})

x^{5} + 1

10 y^{^{''}} + 1000 y = 170 cos (8 t)

\frac{\partial}{\partial x} (e^{x^{2} + 3 y^{2}})

What is the integral of 2/3 e^{3x}x ?
The integral of 2/3 e^{3x}x is 2/27 (3e^{3x}x-e^{3x})+C