What is the integral of xe^{7x} ?

The integral of xe^{7x} is 1/49 (e^{7x}*7x-e^{7x})+C

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integral of xe^{7x}

\int x e^{7 x} d x

\frac{1}{49} (e^{7 x} \cdot 7 x - e^{7 x}) + C

Solution steps

\int x e^{7 x} d x

Apply u-substitution

: \int \frac{e ^{u} u}{49} d u

= \int \frac{e ^{u} u}{49} d u

Take the constant out:

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

= \frac{1}{49} \cdot \int e^{u} u d u

Apply Integration By Parts

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

= \frac{1}{49} (e^{u} u - \int e^{u} d u)

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

= \frac{1}{49} (e^{u} u - e^{u})

Substitute back

u = 7 x

= \frac{1}{49} (e^{7 x} \cdot 7 x - e^{7 x})

Add a constant to the solution

= \frac{1}{49} (e^{7 x} \cdot 7 x - e^{7 x}) + C

x^{2} + y^{2} = 25, y = 0

\int_{0}^{2} 3^{x} d x

arctan (2 x)

\frac{\partial}{\partial y} (\frac{x}{x ^{3} - y ^{7}})

\int \frac{( x ^{2} + x - 1 )}{x ^{2} - x} d x

What is the integral of xe^{7x} ?
The integral of xe^{7x} is 1/49 (e^{7x}*7x-e^{7x})+C