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1/(e^(2*x))dx

Solving integrals/ 1/(e^(2*x))

Integral of 1/(e^(2*x)) dx

The solution

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

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

Detail solution

Let $u = e^{2 x}$ .

Then let $du = 2 e^{2 x} dx$ and substitute $\frac{du}{2}$ :

$\int \frac{1}{2 u^{2}}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u^{2}}\, du = \frac{\int \frac{1}{u^{2}}\, du}{2}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
  So, the result is: $- \frac{1}{2 u}$
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
 |  1            e    
 | ---- dx = C - -----
 |  2*x            2  
 | E                  
 |                    
/

$$\int \frac{1}{e^{2 x}}\, dx = C - \frac{e^{- 2 x}}{2}$$

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

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

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Identical expressions

Integral of 1/(e^(2*x)) dx

Limits of integration:

The graph:

Piecewise:

The solution