What is the integral of x*e^{6x} ?

The integral of x*e^{6x} is 1/36 (e^{6x}*6x-e^{6x})+C

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integral of x*e^{6x}

\int x \cdot e^{6 x} d x

\frac{1}{36} (e^{6 x} \cdot 6 x - e^{6 x}) + C

Solution steps

\int x e^{6 x} d x

Apply u-substitution

= \int \frac{e ^{u} u}{36} d u

Take the constant out:

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

= \frac{1}{36} \cdot \int e^{u} u d u

Apply Integration By Parts

= \frac{1}{36} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{36} (e^{u} u - e^{u})

Substitute back

u = 6 x

= \frac{1}{36} (e^{6 x} \cdot 6 x - e^{6 x})

Add a constant to the solution

= \frac{1}{36} (e^{6 x} \cdot 6 x - e^{6 x}) + C

What is the integral of x*e^{6x} ?
The integral of x*e^{6x} is 1/36 (e^{6x}*6x-e^{6x})+C