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Solving integrals/ -xexp(x)

Integral of -xexp(x) dx

The solution

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  x         
  /         
 |          
 |      x   
 |  -x*e  dx
 |          
/           
0

$$\int\limits_{0}^{x} - x e^{x}\, dx$$

Integral((-x)*exp(x), (x, 0, x))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = - x$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$ .

Then $\operatorname{du}{\left(x \right)} = -1$ .

To find $v{\left(x \right)}$ :
1. The integral of the exponential function is itself.
  
  $\int e^{x}\, dx = e^{x}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- e^{x}\right)\, dx = - \int e^{x}\, dx$
1. The integral of the exponential function is itself.
  
  $\int e^{x}\, dx = e^{x}$
So, the result is: $- e^{x}$
Now simplify:

$\left(1 - x\right) e^{x}$
Add the constant of integration:

$\left(1 - x\right) e^{x}+ \mathrm{constant}$

The answer is:

$\left(1 - x\right) e^{x}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                        
 |                         
 |     x             x    x
 | -x*e  dx = C - x*e  + e 
 |                         
/

$$\int - x e^{x}\, dx = C - x e^{x} + e^{x}$$

Simplify

The answer [src]

              x
-1 + (1 - x)*e

$$\left(1 - x\right) e^{x} - 1$$

=

              x
-1 + (1 - x)*e

$$\left(1 - x\right) e^{x} - 1$$

-1 + (1 - x)*exp(x)

Use the examples entering the upper and lower limits of integration.