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Graphing y =:
exp(-sqrt(x))
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exp(-sqrt(x))
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exp(sqrt(x))
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Integral of exp(-sqrt(x)) dx

The solution

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 oo           
  /           
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 |      ___   
 |   -\/ x    
 |  e       dx
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/             
0

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

Integral(exp(-sqrt(x)), (x, 0, oo))

Detail solution

Let $u = - \sqrt{x}$ .

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

$\int 2 u e^{u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int u e^{u}\, du = 2 \int u e^{u}\, du$
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
    
    Then $\operatorname{du}{\left(u \right)} = 1$ .
    
    To find $v{\left(u \right)}$ :
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    Now evaluate the sub-integral.
  2. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  So, the result is: $2 u e^{u} - 2 e^{u}$
Now substitute $u$ back in:

$- 2 \sqrt{x} e^{- \sqrt{x}} - 2 e^{- \sqrt{x}}$
Now simplify:

$- \left(2 \sqrt{x} + 2\right) e^{- \sqrt{x}}$
Add the constant of integration:

$- \left(2 \sqrt{x} + 2\right) e^{- \sqrt{x}}+ \mathrm{constant}$

The answer is:

$- \left(2 \sqrt{x} + 2\right) e^{- \sqrt{x}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                            
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 |     ___                ___               ___
 |  -\/ x              -\/ x        ___  -\/ x 
 | e       dx = C - 2*e       - 2*\/ x *e      
 |                                             
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$2$$

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

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Identical expressions

Similar expressions

Integral of exp(-sqrt(x)) dx

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The graph:

Piecewise:

The solution