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Identical expressions
-4x*exp(4x)
minus 4x multiply by exponent of (4x)
-4xexp(4x)
-4xexp4x
-4x*exp(4x)dx
Similar expressions
4x*exp(4x)

Integral of -4x*exp(4x) dx

The solution

You have entered [src]

  1             
  /             
 |              
 |        4*x   
 |  -4*x*e    dx
 |              
/               
0

$$\int\limits_{0}^{1} - 4 x e^{4 x}\, dx$$

Integral((-4*x)*exp(4*x), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = - 4 x$ and let $\operatorname{dv}{\left(x \right)} = e^{4 x}$ .

Then $\operatorname{du}{\left(x \right)} = -4$ .

To find $v{\left(x \right)}$ :
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{e^{u}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{4}$
  Now substitute $u$ back in:
  
  $\frac{e^{4 x}}{4}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- e^{4 x}\right)\, dx = - \int e^{4 x}\, dx$
1. Let $u = 4 x$ .
  
  Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :
  
  $\int \frac{e^{u}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{4}$
  Now substitute $u$ back in:
  
  $\frac{e^{4 x}}{4}$
So, the result is: $- \frac{e^{4 x}}{4}$
Now simplify:

$\left(\frac{1}{4} - x\right) e^{4 x}$
Add the constant of integration:

$\left(\frac{1}{4} - x\right) e^{4 x}+ \mathrm{constant}$

The answer is:

$\left(\frac{1}{4} - x\right) e^{4 x}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                     4*x         
 |       4*x          e         4*x
 | -4*x*e    dx = C + ---- - x*e   
 |                     4           
/

$$\int - 4 x e^{4 x}\, dx = C - x e^{4 x} + \frac{e^{4 x}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         4
  1   3*e 
- - - ----
  4    4

$$- \frac{3 e^{4}}{4} - \frac{1}{4}$$

         4
  1   3*e 
- - - ----
  4    4

$$- \frac{3 e^{4}}{4} - \frac{1}{4}$$

-1/4 - 3*exp(4)/4

Numerical answer [src]

-41.1986125248582

-41.1986125248582

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

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Identical expressions

Similar expressions

Integral of -4x*exp(4x) dx

Limits of integration:

The graph:

Piecewise:

The solution