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Integral of d{x}:
Integral of x^(-1/2)
Integral of sin²xcos²x
Integral of x*2^(-x)
Integral of y^(-2/3)
Derivative of:
e^(-2*x)
Limit of the function:
e^(-2*x)
Identical expressions
e^(- two *x)
e to the power of ( minus 2 multiply by x)
e to the power of ( minus two multiply by x)
e(-2*x)
e-2*x
e^(-2x)
e(-2x)
e-2x
e^-2x
e^(-2*x)dx
Similar expressions
e^(2*x)
e^-x/2e^(-2x)
x^2e^(-2x)
1/(1-e^(-2x))
(x-3)e^(-2x)
-2e^(-2x)

Solving integrals/ e^(-2*x)

Integral of e^(-2*x) dx

The solution

You have entered [src]

  1         
  /         
 |          
 |   -2*x   
 |  E     dx
 |          
/           
0

$$\int\limits_{0}^{1} e^{- 2 x}\, dx$$

Integral(E^(-2*x), (x, 0, 1))

Detail solution

Let $u = - 2 x$ .

Then let $du = - 2 dx$ and substitute $- \frac{du}{2}$ :

$\int \left(- \frac{e^{u}}{2}\right)\, du$
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: $- \frac{e^{u}}{2}$
Now substitute $u$ back in:

$- \frac{e^{- 2 x}}{2}$
Add the constant of integration:

$- \frac{e^{- 2 x}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{e^{- 2 x}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                    
 |                 -2*x
 |  -2*x          e    
 | E     dx = C - -----
 |                  2  
/

$$\int e^{- 2 x}\, dx = C - \frac{e^{- 2 x}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

     -2
1   e  
- - ---
2    2

$$\frac{1}{2} - \frac{1}{2 e^{2}}$$

=

     -2
1   e  
- - ---
2    2

$$\frac{1}{2} - \frac{1}{2 e^{2}}$$

1/2 - exp(-2)/2

Numerical answer [src]

0.432332358381694

0.432332358381694

The graph

Integral of e^(-2*x) dx

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