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Solving integrals/ 2^-x

Integral of 2^-x dx

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

Integral(2^(-x), (x, 0, oo))

Detail solution

Let $u = - x$ .

Then let $du = - dx$ and substitute $- du$ :

$\int \left(- 2^{u}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2^{u}\, du = - \int 2^{u}\, du$
  1. The integral of an exponential function is itself divided by the natural logarithm of the base.
    
    $\int 2^{u}\, du = \frac{2^{u}}{\log{\left(2 \right)}}$
  So, the result is: $- \frac{2^{u}}{\log{\left(2 \right)}}$
Now substitute $u$ back in:

$- \frac{2^{- x}}{\log{\left(2 \right)}}$
Add the constant of integration:

$- \frac{2^{- x}}{\log{\left(2 \right)}}+ \mathrm{constant}$

The answer is:

$- \frac{2^{- x}}{\log{\left(2 \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                   
 |                -x  
 |  -x           2    
 | 2   dx = C - ------
 |              log(2)
/

$$\int 2^{- x}\, dx = C - \frac{2^{- x}}{\log{\left(2 \right)}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1   
------
log(2)

$$\frac{1}{\log{\left(2 \right)}}$$

  1   
------
log(2)

$$\frac{1}{\log{\left(2 \right)}}$$

1/log(2)

The graph

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