What is the integral of (xe^{5x}) ?

The integral of (xe^{5x}) is 1/25 (e^{5x}*5x-e^{5x})+C

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integral of (xe^{5x})

\int (x e^{5 x}) d x

\frac{1}{25} (e^{5 x} \cdot 5 x - e^{5 x}) + C

Solution steps

\int x e^{5 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{25} d u

= \int \frac{e ^{u} u}{25} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= \frac{1}{25} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{25} (e^{u} u - e^{u})

Substitute back

u = 5 x

= \frac{1}{25} (e^{5 x} \cdot 5 x - e^{5 x})

Add a constant to the solution

= \frac{1}{25} (e^{5 x} \cdot 5 x - e^{5 x}) + C

\frac{d y}{e ^{- 2 y}} = 8 x^{3} d x

\frac{\partial}{\partial y} (9 x^{\frac{y}{z}})

\frac{d}{d x} (\frac{sin ( 3 x )}{cos ( 2 x )})

arcsin (\frac{x}{9})

f (x) = \frac{e ^{x}}{x}, e, at x = 1

What is the integral of (xe^{5x}) ?
The integral of (xe^{5x}) is 1/25 (e^{5x}*5x-e^{5x})+C