What is the integral of xe^{mx} ?

The integral of xe^{mx} is 1/(m^2)(e^{mx}mx-e^{mx})+C

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integral of xe^{mx}

\int x e^{m x} d x

\frac{1}{m ^{2}} (e^{m x} m x - e^{m x}) + C

Solution steps

\int x e^{m x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{m ^{2}} d u

= \int \frac{e ^{u} u}{m ^{2}} d u

Take the constant out:

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

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

Apply Integration By Parts

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

= \frac{1}{m ^{2}} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{m ^{2}} (e^{u} u - e^{u})

Substitute back

u = m x

= \frac{1}{m ^{2}} (e^{m x} m x - e^{m x})

Add a constant to the solution

= \frac{1}{m ^{2}} (e^{m x} m x - e^{m x}) + C

x \to 0 lim (\frac{e ^{5 x} - 1}{x})

\int cos (5 x) sin (x) sin (8 x) d x

\frac{d y}{d x} = \frac{x cos ( x ^{2} )}{4 y}

\frac{d}{d x} (cos^{2} (x) + cos (x^{2}))

y^{^{'}} = - \frac{x}{4 y}

What is the integral of xe^{mx} ?
The integral of xe^{mx} is 1/(m^2)(e^{mx}mx-e^{mx})+C